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\centerline
{\bf A list of Anomalous Cancellations }
\bigskip
{\it Shalosh B. EKHAD}



                   Lots and Lots of Anomalous Cancellations

                              By Shalosh B. Ekhad

In this  book, you will find listed all the anomalous cancellation where the    denominator has up to , 3, digits
and one digit in the numerator and one in the denominator, that must be the same, gets cancelled, and you still get something correct!

Chapter 1 : The denominator has 2 digits



Section 1

All the Anomalous Cancellations with the denominator with, 2,

    digits, and where the , 1, -th digit of the numerator

          is cancelled out with the, 0,  -th digit of the denominator

denominator= 26


                      $\frac{\not65}{2\not6}=\frac{5}{2}$







Section 2

All the Anomalous Cancellations with the denominator with, 2,

    digits, and where the , 1, -th digit of the numerator

          is cancelled out with the, 1,  -th digit of the denominator

denominator= 64


                     $\frac{1\not60}{\not64}=\frac{10}{4}$



   -------------------------------------------------------------------------

Chapter 2 : The denominator has 3 digits



Section 1

All the Anomalous Cancellations with the denominator with, 3,

    digits, and where the , 1, -th digit of the numerator

          is cancelled out with the, 0,  -th digit of the denominator

denominator= 106


                    $\frac{2\not65}{10\not6}=\frac{25}{10}$



denominator= 146


                    $\frac{3\not65}{14\not6}=\frac{35}{14}$



denominator= 186


                    $\frac{4\not65}{18\not6}=\frac{45}{18}$



denominator= 217


                    $\frac{7\not75}{21\not7}=\frac{75}{21}$



denominator= 222


                   $\frac{12\not21}{22\not2}=\frac{121}{22}$



denominator= 226


                    $\frac{5\not65}{22\not6}=\frac{55}{22}$



denominator= 244


                   $\frac{13\not42}{24\not4}=\frac{132}{24}$



denominator= 248


                   $\frac{10\not85}{24\not8}=\frac{105}{24}$



denominator= 266


                    $\frac{6\not65}{26\not6}=\frac{65}{26}$

                   $\frac{14\not63}{26\not6}=\frac{143}{26}$

                   $\frac{22\not61}{26\not6}=\frac{221}{26}$



denominator= 288


                   $\frac{15\not84}{28\not8}=\frac{154}{28}$



denominator= 305


                    $\frac{8\not54}{30\not5}=\frac{84}{30}$

                   $\frac{19\not52}{30\not5}=\frac{192}{30}$



denominator= 306


                    $\frac{7\not65}{30\not6}=\frac{75}{30}$



denominator= 346


                    $\frac{8\not65}{34\not6}=\frac{85}{34}$



denominator= 357


                   $\frac{12\not75}{35\not7}=\frac{125}{35}$



denominator= 386


                    $\frac{9\not65}{38\not6}=\frac{95}{38}$



denominator= 408


                   $\frac{17\not85}{40\not8}=\frac{175}{40}$



denominator= 422


                   $\frac{23\not21}{42\not2}=\frac{231}{42}$



denominator= 426


                   $\frac{10\not65}{42\not6}=\frac{105}{42}$



denominator= 427


                    $\frac{9\not76}{42\not7}=\frac{96}{42}$

                   $\frac{20\not74}{42\not7}=\frac{204}{42}$

                   $\frac{31\not72}{42\not7}=\frac{312}{42}$



denominator= 444


                   $\frac{14\not43}{44\not4}=\frac{143}{44}$

                   $\frac{24\not42}{44\not4}=\frac{242}{44}$

                   $\frac{34\not41}{44\not4}=\frac{341}{44}$



denominator= 466


                   $\frac{11\not65}{46\not6}=\frac{115}{46}$

                   $\frac{25\not63}{46\not6}=\frac{253}{46}$

                   $\frac{39\not61}{46\not6}=\frac{391}{46}$



denominator= 488


                   $\frac{15\not86}{48\not8}=\frac{156}{48}$

                   $\frac{26\not84}{48\not8}=\frac{264}{48}$

                   $\frac{37\not82}{48\not8}=\frac{372}{48}$



denominator= 497


                   $\frac{17\not75}{49\not7}=\frac{175}{49}$



denominator= 506


                   $\frac{12\not65}{50\not6}=\frac{125}{50}$



denominator= 546


                   $\frac{13\not65}{54\not6}=\frac{135}{54}$



denominator= 555


                   $\frac{15\not54}{55\not5}=\frac{154}{55}$

                   $\frac{25\not53}{55\not5}=\frac{253}{55}$

                   $\frac{35\not52}{55\not5}=\frac{352}{55}$

                   $\frac{45\not51}{55\not5}=\frac{451}{55}$



denominator= 568


                   $\frac{24\not85}{56\not8}=\frac{245}{56}$



denominator= 586


                   $\frac{14\not65}{58\not6}=\frac{145}{58}$



denominator= 622


                   $\frac{34\not21}{62\not2}=\frac{341}{62}$



denominator= 626


                   $\frac{15\not65}{62\not6}=\frac{155}{62}$



denominator= 637


                   $\frac{22\not75}{63\not7}=\frac{225}{63}$



denominator= 644


                   $\frac{35\not42}{64\not4}=\frac{352}{64}$



denominator= 666


                   $\frac{16\not65}{66\not6}=\frac{165}{66}$

                   $\frac{36\not63}{66\not6}=\frac{363}{66}$

                   $\frac{56\not61}{66\not6}=\frac{561}{66}$



denominator= 688


                   $\frac{37\not84}{68\not8}=\frac{374}{68}$



denominator= 706


                   $\frac{17\not65}{70\not6}=\frac{175}{70}$



denominator= 728


                   $\frac{31\not85}{72\not8}=\frac{315}{72}$



denominator= 746


                   $\frac{18\not65}{74\not6}=\frac{185}{74}$



denominator= 777


                   $\frac{17\not76}{77\not7}=\frac{176}{77}$

                   $\frac{27\not75}{77\not7}=\frac{275}{77}$

                   $\frac{37\not74}{77\not7}=\frac{374}{77}$

                   $\frac{47\not73}{77\not7}=\frac{473}{77}$

                   $\frac{57\not72}{77\not7}=\frac{572}{77}$

                   $\frac{67\not71}{77\not7}=\frac{671}{77}$



denominator= 786


                   $\frac{19\not65}{78\not6}=\frac{195}{78}$



denominator= 805


                   $\frac{22\not54}{80\not5}=\frac{224}{80}$

                   $\frac{51\not52}{80\not5}=\frac{512}{80}$



denominator= 822


                   $\frac{45\not21}{82\not2}=\frac{451}{82}$



denominator= 826


                   $\frac{20\not65}{82\not6}=\frac{205}{82}$



denominator= 844


                   $\frac{27\not43}{84\not4}=\frac{273}{84}$

                   $\frac{46\not42}{84\not4}=\frac{462}{84}$

                   $\frac{65\not41}{84\not4}=\frac{651}{84}$



denominator= 866


                   $\frac{21\not65}{86\not6}=\frac{215}{86}$

                   $\frac{47\not63}{86\not6}=\frac{473}{86}$

                   $\frac{73\not61}{86\not6}=\frac{731}{86}$



denominator= 888


                   $\frac{18\not87}{88\not8}=\frac{187}{88}$

                   $\frac{28\not86}{88\not8}=\frac{286}{88}$

                   $\frac{38\not85}{88\not8}=\frac{385}{88}$

                   $\frac{48\not84}{88\not8}=\frac{484}{88}$

                   $\frac{58\not83}{88\not8}=\frac{583}{88}$

                   $\frac{68\not82}{88\not8}=\frac{682}{88}$

                   $\frac{78\not81}{88\not8}=\frac{781}{88}$



denominator= 906


                   $\frac{22\not65}{90\not6}=\frac{225}{90}$



denominator= 917


                   $\frac{32\not75}{91\not7}=\frac{325}{91}$



denominator= 946


                   $\frac{23\not65}{94\not6}=\frac{235}{94}$



denominator= 986


                   $\frac{24\not65}{98\not6}=\frac{245}{98}$



   -------------------------------------------------------------------------



Section 2

All the Anomalous Cancellations with the denominator with, 3,

    digits, and where the , 2, -th digit of the numerator

          is cancelled out with the, 0,  -th digit of the denominator

denominator= 106


                   $\frac{6\not625}{10\not6}=\frac{625}{10}$



denominator= 124


                   $\frac{5\not425}{12\not4}=\frac{525}{12}$



denominator= 186


                  $\frac{11\not625}{18\not6}=\frac{1125}{18}$



denominator= 217


                    $\frac{\not775}{21\not7}=\frac{75}{21}$

                   $\frac{7\not750}{21\not7}=\frac{750}{21}$

                  $\frac{14\not725}{21\not7}=\frac{1425}{21}$



denominator= 248


                   $\frac{3\not875}{24\not8}=\frac{375}{24}$

                  $\frac{10\not850}{24\not8}=\frac{1050}{24}$

                  $\frac{17\not825}{24\not8}=\frac{1725}{24}$



denominator= 266


                    $\frac{\not665}{26\not6}=\frac{65}{26}$

                   $\frac{4\not655}{26\not6}=\frac{455}{26}$

                   $\frac{8\not645}{26\not6}=\frac{845}{26}$

                  $\frac{12\not635}{26\not6}=\frac{1235}{26}$

                  $\frac{16\not625}{26\not6}=\frac{1625}{26}$

                  $\frac{20\not615}{26\not6}=\frac{2015}{26}$

                  $\frac{24\not605}{26\not6}=\frac{2405}{26}$



denominator= 284


                  $\frac{12\not425}{28\not4}=\frac{1225}{28}$



denominator= 346


                  $\frac{21\not625}{34\not6}=\frac{2125}{34}$



denominator= 357


                  $\frac{12\not750}{35\not7}=\frac{1250}{35}$



denominator= 408


                  $\frac{17\not850}{40\not8}=\frac{1750}{40}$



denominator= 426


                  $\frac{26\not625}{42\not6}=\frac{2625}{42}$



denominator= 427


                   $\frac{9\not760}{42\not7}=\frac{960}{42}$

                  $\frac{20\not740}{42\not7}=\frac{2040}{42}$

                  $\frac{31\not720}{42\not7}=\frac{3120}{42}$



denominator= 488


                  $\frac{15\not860}{48\not8}=\frac{1560}{48}$

                  $\frac{37\not820}{48\not8}=\frac{3720}{48}$



denominator= 497


                   $\frac{1\not775}{49\not7}=\frac{175}{49}$

                  $\frac{17\not750}{49\not7}=\frac{1750}{49}$

                  $\frac{33\not725}{49\not7}=\frac{3325}{49}$



denominator= 506


                  $\frac{31\not625}{50\not6}=\frac{3125}{50}$



denominator= 568


                   $\frac{8\not875}{56\not8}=\frac{875}{56}$

                  $\frac{24\not850}{56\not8}=\frac{2450}{56}$

                  $\frac{40\not825}{56\not8}=\frac{4025}{56}$



denominator= 586


                  $\frac{36\not625}{58\not6}=\frac{3625}{58}$



denominator= 604


                  $\frac{26\not425}{60\not4}=\frac{2625}{60}$



denominator= 622


                  $\frac{20\not215}{62\not2}=\frac{2015}{62}$

                  $\frac{48\not205}{62\not2}=\frac{4805}{62}$



denominator= 637


                  $\frac{22\not750}{63\not7}=\frac{2250}{63}$



denominator= 728


                  $\frac{31\not850}{72\not8}=\frac{3150}{72}$



denominator= 746


                  $\frac{46\not625}{74\not6}=\frac{4625}{74}$



denominator= 764


                  $\frac{33\not425}{76\not4}=\frac{3325}{76}$



denominator= 826


                  $\frac{51\not625}{82\not6}=\frac{5125}{82}$



denominator= 844


                  $\frac{27\not430}{84\not4}=\frac{2730}{84}$

                  $\frac{65\not410}{84\not4}=\frac{6510}{84}$



denominator= 906


                  $\frac{56\not625}{90\not6}=\frac{5625}{90}$



denominator= 917


                  $\frac{32\not750}{91\not7}=\frac{3250}{91}$



denominator= 924


                  $\frac{40\not425}{92\not4}=\frac{4025}{92}$



denominator= 986


                  $\frac{61\not625}{98\not6}=\frac{6125}{98}$



   -------------------------------------------------------------------------



Section 3

All the Anomalous Cancellations with the denominator with, 3,

    digits, and where the , 1, -th digit of the numerator

          is cancelled out with the, 1,  -th digit of the denominator

denominator= 160


                     $\frac{\not64}{1\not60}=\frac{4}{10}$



denominator= 190


                     $\frac{\not95}{1\not90}=\frac{5}{10}$



denominator= 192


                     $\frac{\not96}{1\not92}=\frac{6}{12}$



denominator= 194


                     $\frac{\not97}{1\not94}=\frac{7}{14}$

                    $\frac{2\not91}{1\not94}=\frac{21}{14}$



denominator= 196


                     $\frac{\not98}{1\not96}=\frac{8}{16}$

                    $\frac{2\not94}{1\not96}=\frac{24}{16}$

                    $\frac{4\not90}{1\not96}=\frac{40}{16}$



denominator= 222


                   $\frac{12\not21}{2\not22}=\frac{121}{22}$



denominator= 244


                   $\frac{13\not42}{2\not44}=\frac{132}{24}$



denominator= 260


                     $\frac{\not65}{2\not60}=\frac{5}{20}$



denominator= 266


                    $\frac{6\not65}{2\not66}=\frac{65}{26}$

                   $\frac{14\not63}{2\not66}=\frac{143}{26}$

                   $\frac{22\not61}{2\not66}=\frac{221}{26}$



denominator= 288


                   $\frac{15\not84}{2\not88}=\frac{154}{28}$



denominator= 291


                     $\frac{\not97}{2\not91}=\frac{7}{21}$

                    $\frac{1\not94}{2\not91}=\frac{14}{21}$



denominator= 294


                     $\frac{\not98}{2\not94}=\frac{8}{24}$

                    $\frac{1\not96}{2\not94}=\frac{16}{24}$

                    $\frac{3\not92}{2\not94}=\frac{32}{24}$

                    $\frac{4\not90}{2\not94}=\frac{40}{24}$



denominator= 332


                    $\frac{8\not30}{3\not32}=\frac{80}{32}$



denominator= 390


                    $\frac{1\not95}{3\not90}=\frac{15}{30}$



denominator= 392


                     $\frac{\not98}{3\not92}=\frac{8}{32}$

                    $\frac{1\not96}{3\not92}=\frac{16}{32}$

                    $\frac{2\not94}{3\not92}=\frac{24}{32}$

                    $\frac{4\not90}{3\not92}=\frac{40}{32}$



denominator= 394


                    $\frac{1\not97}{3\not94}=\frac{17}{34}$

                    $\frac{5\not91}{3\not94}=\frac{51}{34}$



denominator= 398


                    $\frac{1\not99}{3\not98}=\frac{19}{38}$

                    $\frac{5\not97}{3\not98}=\frac{57}{38}$

                    $\frac{9\not95}{3\not98}=\frac{95}{38}$

                   $\frac{13\not93}{3\not98}=\frac{133}{38}$

                   $\frac{17\not91}{3\not98}=\frac{171}{38}$



denominator= 422


                   $\frac{23\not21}{4\not22}=\frac{231}{42}$



denominator= 444


                   $\frac{14\not43}{4\not44}=\frac{143}{44}$

                   $\frac{24\not42}{4\not44}=\frac{242}{44}$

                   $\frac{34\not41}{4\not44}=\frac{341}{44}$



denominator= 464


                   $\frac{11\not60}{4\not64}=\frac{110}{44}$



denominator= 466


                   $\frac{11\not65}{4\not66}=\frac{115}{46}$

                   $\frac{25\not63}{4\not66}=\frac{253}{46}$

                   $\frac{39\not61}{4\not66}=\frac{391}{46}$



denominator= 488


                   $\frac{15\not86}{4\not88}=\frac{156}{48}$

                   $\frac{26\not84}{4\not88}=\frac{264}{48}$

                   $\frac{37\not82}{4\not88}=\frac{372}{48}$



denominator= 490


                     $\frac{\not98}{4\not90}=\frac{8}{40}$

                    $\frac{1\not96}{4\not90}=\frac{16}{40}$

                    $\frac{2\not94}{4\not90}=\frac{24}{40}$

                    $\frac{3\not92}{4\not90}=\frac{32}{40}$



denominator= 532


                    $\frac{1\not33}{5\not32}=\frac{13}{52}$

                    $\frac{9\not31}{5\not32}=\frac{91}{52}$

                   $\frac{13\not30}{5\not32}=\frac{130}{52}$



denominator= 555


                   $\frac{15\not54}{5\not55}=\frac{154}{55}$

                   $\frac{25\not53}{5\not55}=\frac{253}{55}$

                   $\frac{35\not52}{5\not55}=\frac{352}{55}$

                   $\frac{45\not51}{5\not55}=\frac{451}{55}$



denominator= 576


                   $\frac{18\not72}{5\not76}=\frac{182}{56}$



denominator= 590


                    $\frac{2\not95}{5\not90}=\frac{25}{50}$



denominator= 591


                    $\frac{1\not97}{5\not91}=\frac{17}{51}$

                    $\frac{3\not94}{5\not91}=\frac{34}{51}$



denominator= 592


                    $\frac{2\not96}{5\not92}=\frac{26}{52}$



denominator= 596


                    $\frac{2\not98}{5\not96}=\frac{28}{56}$

                    $\frac{8\not94}{5\not96}=\frac{84}{56}$

                   $\frac{14\not90}{5\not96}=\frac{140}{56}$



denominator= 597


                    $\frac{1\not99}{5\not97}=\frac{19}{57}$

                    $\frac{3\not98}{5\not97}=\frac{38}{57}$

                    $\frac{7\not96}{5\not97}=\frac{76}{57}$

                    $\frac{9\not95}{5\not97}=\frac{95}{57}$

                   $\frac{13\not93}{5\not97}=\frac{133}{57}$

                   $\frac{15\not92}{5\not97}=\frac{152}{57}$

                   $\frac{19\not90}{5\not97}=\frac{190}{57}$



denominator= 598


                    $\frac{2\not99}{5\not98}=\frac{29}{58}$

                    $\frac{8\not97}{5\not98}=\frac{87}{58}$

                   $\frac{14\not95}{5\not98}=\frac{145}{58}$

                   $\frac{20\not93}{5\not98}=\frac{203}{58}$

                   $\frac{26\not91}{5\not98}=\frac{261}{58}$



denominator= 622


                   $\frac{34\not21}{6\not22}=\frac{341}{62}$



denominator= 644


                   $\frac{35\not42}{6\not44}=\frac{352}{64}$



denominator= 664


                    $\frac{1\not66}{6\not64}=\frac{16}{64}$

                   $\frac{11\not62}{6\not64}=\frac{112}{64}$

                   $\frac{16\not60}{6\not64}=\frac{160}{64}$



denominator= 665


                    $\frac{2\not66}{6\not65}=\frac{26}{65}$

                   $\frac{10\not64}{6\not65}=\frac{104}{65}$

                   $\frac{14\not63}{6\not65}=\frac{143}{65}$

                   $\frac{18\not62}{6\not65}=\frac{182}{65}$

                   $\frac{22\not61}{6\not65}=\frac{221}{65}$



denominator= 666


                   $\frac{16\not65}{6\not66}=\frac{165}{66}$

                   $\frac{36\not63}{6\not66}=\frac{363}{66}$

                   $\frac{56\not61}{6\not66}=\frac{561}{66}$



denominator= 688


                   $\frac{37\not84}{6\not88}=\frac{374}{68}$



denominator= 732


                   $\frac{18\not30}{7\not32}=\frac{180}{72}$



denominator= 775


                   $\frac{21\not70}{7\not75}=\frac{210}{75}$



denominator= 777


                   $\frac{17\not76}{7\not77}=\frac{176}{77}$

                   $\frac{27\not75}{7\not77}=\frac{275}{77}$

                   $\frac{37\not74}{7\not77}=\frac{374}{77}$

                   $\frac{47\not73}{7\not77}=\frac{473}{77}$

                   $\frac{57\not72}{7\not77}=\frac{572}{77}$

                   $\frac{67\not71}{7\not77}=\frac{671}{77}$



denominator= 790


                    $\frac{3\not95}{7\not90}=\frac{35}{70}$



denominator= 794


                    $\frac{3\not97}{7\not94}=\frac{37}{74}$

                   $\frac{11\not91}{7\not94}=\frac{111}{74}$



denominator= 796


                    $\frac{1\not99}{7\not96}=\frac{19}{76}$

                    $\frac{3\not98}{7\not96}=\frac{38}{76}$

                    $\frac{5\not97}{7\not96}=\frac{57}{76}$

                    $\frac{9\not95}{7\not96}=\frac{95}{76}$

                   $\frac{11\not94}{7\not96}=\frac{114}{76}$

                   $\frac{13\not93}{7\not96}=\frac{133}{76}$

                   $\frac{17\not91}{7\not96}=\frac{171}{76}$

                   $\frac{19\not90}{7\not96}=\frac{190}{76}$



denominator= 798


                    $\frac{3\not99}{7\not98}=\frac{39}{78}$

                   $\frac{11\not97}{7\not98}=\frac{117}{78}$

                   $\frac{19\not95}{7\not98}=\frac{195}{78}$

                   $\frac{27\not93}{7\not98}=\frac{273}{78}$

                   $\frac{35\not91}{7\not98}=\frac{351}{78}$



denominator= 822


                   $\frac{45\not21}{8\not22}=\frac{451}{82}$



denominator= 830


                    $\frac{3\not32}{8\not30}=\frac{32}{80}$



denominator= 844


                   $\frac{27\not43}{8\not44}=\frac{273}{84}$

                   $\frac{46\not42}{8\not44}=\frac{462}{84}$

                   $\frac{65\not41}{8\not44}=\frac{651}{84}$



denominator= 854


                   $\frac{19\not52}{8\not54}=\frac{192}{84}$

                   $\frac{30\not50}{8\not54}=\frac{300}{84}$



denominator= 864


                   $\frac{21\not60}{8\not64}=\frac{210}{84}$



denominator= 866


                   $\frac{21\not65}{8\not66}=\frac{215}{86}$

                   $\frac{47\not63}{8\not66}=\frac{473}{86}$

                   $\frac{73\not61}{8\not66}=\frac{731}{86}$



denominator= 888


                   $\frac{18\not87}{8\not88}=\frac{187}{88}$

                   $\frac{28\not86}{8\not88}=\frac{286}{88}$

                   $\frac{38\not85}{8\not88}=\frac{385}{88}$

                   $\frac{48\not84}{8\not88}=\frac{484}{88}$

                   $\frac{58\not83}{8\not88}=\frac{583}{88}$

                   $\frac{68\not82}{8\not88}=\frac{682}{88}$

                   $\frac{78\not81}{8\not88}=\frac{781}{88}$



denominator= 894


                    $\frac{2\not98}{8\not94}=\frac{28}{84}$

                    $\frac{5\not96}{8\not94}=\frac{56}{84}$

                   $\frac{11\not92}{8\not94}=\frac{112}{84}$

                   $\frac{14\not90}{8\not94}=\frac{140}{84}$



denominator= 897


                    $\frac{2\not99}{8\not97}=\frac{29}{87}$

                    $\frac{5\not98}{8\not97}=\frac{58}{87}$

                   $\frac{11\not96}{8\not97}=\frac{116}{87}$

                   $\frac{14\not95}{8\not97}=\frac{145}{87}$

                   $\frac{20\not93}{8\not97}=\frac{203}{87}$

                   $\frac{23\not92}{8\not97}=\frac{232}{87}$

                   $\frac{29\not90}{8\not97}=\frac{290}{87}$



denominator= 931


                    $\frac{1\not33}{9\not31}=\frac{13}{91}$

                    $\frac{5\not32}{9\not31}=\frac{52}{91}$

                   $\frac{13\not30}{9\not31}=\frac{130}{91}$



denominator= 932


                    $\frac{2\not33}{9\not32}=\frac{23}{92}$

                   $\frac{16\not31}{9\not32}=\frac{161}{92}$

                   $\frac{23\not30}{9\not32}=\frac{230}{92}$



denominator= 976


                   $\frac{20\not74}{9\not76}=\frac{204}{96}$

                   $\frac{31\not72}{9\not76}=\frac{312}{96}$

                   $\frac{42\not70}{9\not76}=\frac{420}{96}$



denominator= 992


                    $\frac{4\not96}{9\not92}=\frac{46}{92}$



denominator= 994


                    $\frac{4\not97}{9\not94}=\frac{47}{94}$

                   $\frac{14\not91}{9\not94}=\frac{141}{94}$



denominator= 995


                    $\frac{1\not99}{9\not95}=\frac{19}{95}$

                    $\frac{3\not98}{9\not95}=\frac{38}{95}$

                    $\frac{5\not97}{9\not95}=\frac{57}{95}$

                    $\frac{7\not96}{9\not95}=\frac{76}{95}$

                   $\frac{11\not94}{9\not95}=\frac{114}{95}$

                   $\frac{13\not93}{9\not95}=\frac{133}{95}$

                   $\frac{15\not92}{9\not95}=\frac{152}{95}$

                   $\frac{17\not91}{9\not95}=\frac{171}{95}$



denominator= 996


                    $\frac{4\not98}{9\not96}=\frac{48}{96}$

                   $\frac{14\not94}{9\not96}=\frac{144}{96}$

                   $\frac{24\not90}{9\not96}=\frac{240}{96}$



denominator= 998


                    $\frac{4\not99}{9\not98}=\frac{49}{98}$

                   $\frac{14\not97}{9\not98}=\frac{147}{98}$

                   $\frac{24\not95}{9\not98}=\frac{245}{98}$

                   $\frac{34\not93}{9\not98}=\frac{343}{98}$

                   $\frac{44\not91}{9\not98}=\frac{441}{98}$



   -------------------------------------------------------------------------



Section 4

All the Anomalous Cancellations with the denominator with, 3,

    digits, and where the , 2, -th digit of the numerator

          is cancelled out with the, 1,  -th digit of the denominator

denominator= 121


                    $\frac{\not220}{1\not21}=\frac{20}{11}$

                   $\frac{2\not200}{1\not21}=\frac{200}{11}$



denominator= 130


                    $\frac{\not325}{1\not30}=\frac{25}{10}$



denominator= 132


                    $\frac{\not330}{1\not32}=\frac{30}{12}$

                   $\frac{2\not310}{1\not32}=\frac{210}{12}$



denominator= 143


                    $\frac{\not440}{1\not43}=\frac{40}{13}$

                   $\frac{2\not420}{1\not43}=\frac{220}{13}$

                   $\frac{3\not410}{1\not43}=\frac{310}{13}$

                   $\frac{4\not400}{1\not43}=\frac{400}{13}$



denominator= 154


                    $\frac{\not550}{1\not54}=\frac{50}{14}$

                   $\frac{2\not530}{1\not54}=\frac{230}{14}$

                   $\frac{3\not520}{1\not54}=\frac{320}{14}$

                   $\frac{4\not510}{1\not54}=\frac{410}{14}$

                   $\frac{5\not500}{1\not54}=\frac{500}{14}$



denominator= 176


                    $\frac{\not770}{1\not76}=\frac{70}{16}$

                   $\frac{2\not750}{1\not76}=\frac{250}{16}$

                   $\frac{3\not740}{1\not76}=\frac{340}{16}$

                   $\frac{4\not730}{1\not76}=\frac{430}{16}$

                   $\frac{5\not720}{1\not76}=\frac{520}{16}$

                   $\frac{6\not710}{1\not76}=\frac{610}{16}$

                   $\frac{7\not700}{1\not76}=\frac{700}{16}$



denominator= 187


                    $\frac{\not880}{1\not87}=\frac{80}{17}$

                   $\frac{2\not860}{1\not87}=\frac{260}{17}$

                   $\frac{3\not850}{1\not87}=\frac{350}{17}$

                   $\frac{4\not840}{1\not87}=\frac{440}{17}$

                   $\frac{5\not830}{1\not87}=\frac{530}{17}$

                   $\frac{6\not820}{1\not87}=\frac{620}{17}$

                   $\frac{7\not810}{1\not87}=\frac{710}{17}$

                   $\frac{8\not800}{1\not87}=\frac{800}{17}$



denominator= 220


                   $\frac{1\not210}{2\not20}=\frac{110}{20}$



denominator= 231


                    $\frac{\not330}{2\not31}=\frac{30}{21}$

                   $\frac{1\not320}{2\not31}=\frac{120}{21}$

                   $\frac{3\not300}{2\not31}=\frac{300}{21}$



denominator= 242


                    $\frac{\not440}{2\not42}=\frac{40}{22}$

                   $\frac{1\not430}{2\not42}=\frac{130}{22}$

                   $\frac{3\not410}{2\not42}=\frac{310}{22}$

                   $\frac{4\not400}{2\not42}=\frac{400}{22}$



denominator= 253


                    $\frac{\not550}{2\not53}=\frac{50}{23}$

                   $\frac{1\not540}{2\not53}=\frac{140}{23}$

                   $\frac{3\not520}{2\not53}=\frac{320}{23}$

                   $\frac{4\not510}{2\not53}=\frac{410}{23}$

                   $\frac{5\not500}{2\not53}=\frac{500}{23}$



denominator= 260


                    $\frac{\not650}{2\not60}=\frac{50}{20}$

                   $\frac{1\not625}{2\not60}=\frac{125}{20}$



denominator= 262


                    $\frac{\not655}{2\not62}=\frac{55}{22}$



denominator= 264


                    $\frac{\not660}{2\not64}=\frac{60}{24}$

                   $\frac{1\not650}{2\not64}=\frac{150}{24}$

                   $\frac{3\not630}{2\not64}=\frac{330}{24}$

                   $\frac{4\not620}{2\not64}=\frac{420}{24}$

                   $\frac{5\not610}{2\not64}=\frac{510}{24}$



denominator= 266


                    $\frac{\not665}{2\not66}=\frac{65}{26}$

                   $\frac{4\not655}{2\not66}=\frac{455}{26}$

                   $\frac{8\not645}{2\not66}=\frac{845}{26}$

                  $\frac{12\not635}{2\not66}=\frac{1235}{26}$

                  $\frac{16\not625}{2\not66}=\frac{1625}{26}$

                  $\frac{20\not615}{2\not66}=\frac{2015}{26}$

                  $\frac{24\not605}{2\not66}=\frac{2405}{26}$



denominator= 270


                    $\frac{\not756}{2\not70}=\frac{56}{20}$

                   $\frac{1\not728}{2\not70}=\frac{128}{20}$



denominator= 275


                    $\frac{\not770}{2\not75}=\frac{70}{25}$

                   $\frac{1\not760}{2\not75}=\frac{160}{25}$

                   $\frac{3\not740}{2\not75}=\frac{340}{25}$

                   $\frac{4\not730}{2\not75}=\frac{430}{25}$

                   $\frac{5\not720}{2\not75}=\frac{520}{25}$

                   $\frac{6\not710}{2\not75}=\frac{610}{25}$



denominator= 286


                    $\frac{\not880}{2\not86}=\frac{80}{26}$

                   $\frac{1\not870}{2\not86}=\frac{170}{26}$

                   $\frac{3\not850}{2\not86}=\frac{350}{26}$

                   $\frac{4\not840}{2\not86}=\frac{440}{26}$

                   $\frac{5\not830}{2\not86}=\frac{530}{26}$

                   $\frac{6\not820}{2\not86}=\frac{620}{26}$

                   $\frac{7\not810}{2\not86}=\frac{710}{26}$

                   $\frac{8\not800}{2\not86}=\frac{800}{26}$



denominator= 291


                    $\frac{\not970}{2\not91}=\frac{70}{21}$

                   $\frac{1\not940}{2\not91}=\frac{140}{21}$



denominator= 294


                    $\frac{\not980}{2\not94}=\frac{80}{24}$

                   $\frac{1\not960}{2\not94}=\frac{160}{24}$

                   $\frac{3\not920}{2\not94}=\frac{320}{24}$

                   $\frac{4\not900}{2\not94}=\frac{400}{24}$



denominator= 297


                    $\frac{\not990}{2\not97}=\frac{90}{27}$

                   $\frac{1\not980}{2\not97}=\frac{180}{27}$

                   $\frac{3\not960}{2\not97}=\frac{360}{27}$

                   $\frac{4\not950}{2\not97}=\frac{450}{27}$

                   $\frac{6\not930}{2\not97}=\frac{630}{27}$

                   $\frac{7\not920}{2\not97}=\frac{720}{27}$

                   $\frac{9\not900}{2\not97}=\frac{900}{27}$



denominator= 341


                    $\frac{\not440}{3\not41}=\frac{40}{31}$

                   $\frac{1\not430}{3\not41}=\frac{130}{31}$

                   $\frac{2\not420}{3\not41}=\frac{220}{31}$

                   $\frac{4\not400}{3\not41}=\frac{400}{31}$



denominator= 352


                    $\frac{\not550}{3\not52}=\frac{50}{32}$

                   $\frac{1\not540}{3\not52}=\frac{140}{32}$

                   $\frac{2\not530}{3\not52}=\frac{230}{32}$

                   $\frac{4\not510}{3\not52}=\frac{410}{32}$

                   $\frac{5\not500}{3\not52}=\frac{500}{32}$



denominator= 354


                  $\frac{11\not505}{3\not54}=\frac{1105}{34}$



denominator= 363


                    $\frac{\not660}{3\not63}=\frac{60}{33}$

                   $\frac{1\not650}{3\not63}=\frac{150}{33}$

                   $\frac{2\not640}{3\not63}=\frac{240}{33}$

                   $\frac{4\not620}{3\not63}=\frac{420}{33}$

                   $\frac{5\not610}{3\not63}=\frac{510}{33}$

                   $\frac{6\not600}{3\not63}=\frac{600}{33}$



denominator= 374


                    $\frac{\not770}{3\not74}=\frac{70}{34}$

                   $\frac{1\not760}{3\not74}=\frac{160}{34}$

                   $\frac{2\not750}{3\not74}=\frac{250}{34}$

                   $\frac{4\not730}{3\not74}=\frac{430}{34}$

                   $\frac{5\not720}{3\not74}=\frac{520}{34}$

                   $\frac{6\not710}{3\not74}=\frac{610}{34}$

                   $\frac{7\not700}{3\not74}=\frac{700}{34}$



denominator= 385


                    $\frac{\not880}{3\not85}=\frac{80}{35}$

                   $\frac{1\not870}{3\not85}=\frac{170}{35}$

                   $\frac{2\not860}{3\not85}=\frac{260}{35}$

                   $\frac{4\not840}{3\not85}=\frac{440}{35}$

                   $\frac{5\not830}{3\not85}=\frac{530}{35}$

                   $\frac{6\not820}{3\not85}=\frac{620}{35}$

                   $\frac{7\not810}{3\not85}=\frac{710}{35}$

                   $\frac{8\not800}{3\not85}=\frac{800}{35}$



denominator= 390


                    $\frac{\not975}{3\not90}=\frac{75}{30}$

                   $\frac{2\not925}{3\not90}=\frac{225}{30}$



denominator= 392


                    $\frac{\not980}{3\not92}=\frac{80}{32}$

                   $\frac{2\not940}{3\not92}=\frac{240}{32}$

                   $\frac{4\not900}{3\not92}=\frac{400}{32}$



denominator= 394


                    $\frac{\not985}{3\not94}=\frac{85}{34}$

                   $\frac{2\not955}{3\not94}=\frac{255}{34}$

                   $\frac{4\not925}{3\not94}=\frac{425}{34}$



denominator= 396


                    $\frac{\not990}{3\not96}=\frac{90}{36}$

                   $\frac{2\not970}{3\not96}=\frac{270}{36}$

                   $\frac{4\not950}{3\not96}=\frac{450}{36}$

                   $\frac{6\not930}{3\not96}=\frac{630}{36}$

                   $\frac{8\not910}{3\not96}=\frac{810}{36}$



denominator= 398


                    $\frac{\not995}{3\not98}=\frac{95}{38}$

                   $\frac{2\not985}{3\not98}=\frac{285}{38}$

                   $\frac{4\not975}{3\not98}=\frac{475}{38}$

                   $\frac{6\not965}{3\not98}=\frac{665}{38}$

                   $\frac{8\not955}{3\not98}=\frac{855}{38}$

                  $\frac{10\not945}{3\not98}=\frac{1045}{38}$

                  $\frac{12\not935}{3\not98}=\frac{1235}{38}$

                  $\frac{14\not925}{3\not98}=\frac{1425}{38}$

                  $\frac{16\not915}{3\not98}=\frac{1615}{38}$

                  $\frac{18\not905}{3\not98}=\frac{1805}{38}$



denominator= 440


                   $\frac{1\not430}{4\not40}=\frac{130}{40}$

                   $\frac{2\not420}{4\not40}=\frac{220}{40}$

                   $\frac{3\not410}{4\not40}=\frac{310}{40}$



denominator= 442


                   $\frac{2\not431}{4\not42}=\frac{231}{42}$

                   $\frac{6\not409}{4\not42}=\frac{609}{42}$



denominator= 451


                    $\frac{\not550}{4\not51}=\frac{50}{41}$

                   $\frac{1\not540}{4\not51}=\frac{140}{41}$

                   $\frac{2\not530}{4\not51}=\frac{230}{41}$

                   $\frac{3\not520}{4\not51}=\frac{320}{41}$

                   $\frac{5\not500}{4\not51}=\frac{500}{41}$



denominator= 462


                    $\frac{\not660}{4\not62}=\frac{60}{42}$

                   $\frac{1\not650}{4\not62}=\frac{150}{42}$

                   $\frac{2\not640}{4\not62}=\frac{240}{42}$

                   $\frac{3\not630}{4\not62}=\frac{330}{42}$

                   $\frac{5\not610}{4\not62}=\frac{510}{42}$

                   $\frac{6\not600}{4\not62}=\frac{600}{42}$



denominator= 473


                    $\frac{\not770}{4\not73}=\frac{70}{43}$

                   $\frac{1\not760}{4\not73}=\frac{160}{43}$

                   $\frac{2\not750}{4\not73}=\frac{250}{43}$

                   $\frac{3\not740}{4\not73}=\frac{340}{43}$

                   $\frac{5\not720}{4\not73}=\frac{520}{43}$

                   $\frac{6\not710}{4\not73}=\frac{610}{43}$

                   $\frac{7\not700}{4\not73}=\frac{700}{43}$



denominator= 484


                    $\frac{\not880}{4\not84}=\frac{80}{44}$

                   $\frac{1\not870}{4\not84}=\frac{170}{44}$

                   $\frac{2\not860}{4\not84}=\frac{260}{44}$

                   $\frac{3\not850}{4\not84}=\frac{350}{44}$

                   $\frac{5\not830}{4\not84}=\frac{530}{44}$

                   $\frac{6\not820}{4\not84}=\frac{620}{44}$

                   $\frac{7\not810}{4\not84}=\frac{710}{44}$

                   $\frac{8\not800}{4\not84}=\frac{800}{44}$



denominator= 488


                  $\frac{15\not860}{4\not88}=\frac{1560}{48}$

                  $\frac{37\not820}{4\not88}=\frac{3720}{48}$



denominator= 530


                   $\frac{1\not325}{5\not30}=\frac{125}{50}$



denominator= 532


                   $\frac{1\not330}{5\not32}=\frac{130}{52}$

                   $\frac{3\not325}{5\not32}=\frac{325}{52}$

                   $\frac{7\not315}{5\not32}=\frac{715}{52}$

                   $\frac{9\not310}{5\not32}=\frac{910}{52}$

                  $\frac{11\not305}{5\not32}=\frac{1105}{52}$



denominator= 550


                   $\frac{1\not540}{5\not50}=\frac{140}{50}$

                   $\frac{2\not530}{5\not50}=\frac{230}{50}$

                   $\frac{3\not520}{5\not50}=\frac{320}{50}$

                   $\frac{4\not510}{5\not50}=\frac{410}{50}$



denominator= 561


                    $\frac{\not660}{5\not61}=\frac{60}{51}$

                   $\frac{1\not650}{5\not61}=\frac{150}{51}$

                   $\frac{2\not640}{5\not61}=\frac{240}{51}$

                   $\frac{3\not630}{5\not61}=\frac{330}{51}$

                   $\frac{4\not620}{5\not61}=\frac{420}{51}$

                   $\frac{6\not600}{5\not61}=\frac{600}{51}$



denominator= 572


                    $\frac{\not770}{5\not72}=\frac{70}{52}$

                   $\frac{1\not760}{5\not72}=\frac{160}{52}$

                   $\frac{2\not750}{5\not72}=\frac{250}{52}$

                   $\frac{3\not740}{5\not72}=\frac{340}{52}$

                   $\frac{4\not730}{5\not72}=\frac{430}{52}$

                   $\frac{6\not710}{5\not72}=\frac{610}{52}$

                   $\frac{7\not700}{5\not72}=\frac{700}{52}$



denominator= 576


                  $\frac{18\not720}{5\not76}=\frac{1820}{56}$



denominator= 583


                    $\frac{\not880}{5\not83}=\frac{80}{53}$

                   $\frac{1\not870}{5\not83}=\frac{170}{53}$

                   $\frac{2\not860}{5\not83}=\frac{260}{53}$

                   $\frac{3\not850}{5\not83}=\frac{350}{53}$

                   $\frac{4\not840}{5\not83}=\frac{440}{53}$

                   $\frac{6\not820}{5\not83}=\frac{620}{53}$

                   $\frac{7\not810}{5\not83}=\frac{710}{53}$

                   $\frac{8\not800}{5\not83}=\frac{800}{53}$



denominator= 591


                    $\frac{\not985}{5\not91}=\frac{85}{51}$

                   $\frac{1\not970}{5\not91}=\frac{170}{51}$

                   $\frac{3\not940}{5\not91}=\frac{340}{51}$

                   $\frac{4\not925}{5\not91}=\frac{425}{51}$



denominator= 594


                    $\frac{\not990}{5\not94}=\frac{90}{54}$

                   $\frac{1\not980}{5\not94}=\frac{180}{54}$

                   $\frac{3\not960}{5\not94}=\frac{360}{54}$

                   $\frac{4\not950}{5\not94}=\frac{450}{54}$

                   $\frac{6\not930}{5\not94}=\frac{630}{54}$

                   $\frac{7\not920}{5\not94}=\frac{720}{54}$

                   $\frac{9\not900}{5\not94}=\frac{900}{54}$



denominator= 597


                    $\frac{\not995}{5\not97}=\frac{95}{57}$

                   $\frac{1\not990}{5\not97}=\frac{190}{57}$

                   $\frac{3\not980}{5\not97}=\frac{380}{57}$

                   $\frac{4\not975}{5\not97}=\frac{475}{57}$

                   $\frac{6\not965}{5\not97}=\frac{665}{57}$

                   $\frac{7\not960}{5\not97}=\frac{760}{57}$

                   $\frac{9\not950}{5\not97}=\frac{950}{57}$

                  $\frac{10\not945}{5\not97}=\frac{1045}{57}$

                  $\frac{12\not935}{5\not97}=\frac{1235}{57}$

                  $\frac{13\not930}{5\not97}=\frac{1330}{57}$

                  $\frac{15\not920}{5\not97}=\frac{1520}{57}$

                  $\frac{16\not915}{5\not97}=\frac{1615}{57}$

                  $\frac{18\not905}{5\not97}=\frac{1805}{57}$

                  $\frac{19\not900}{5\not97}=\frac{1900}{57}$



denominator= 622


                  $\frac{20\not215}{6\not22}=\frac{2015}{62}$

                  $\frac{48\not205}{6\not22}=\frac{4805}{62}$



denominator= 660


                   $\frac{1\not650}{6\not60}=\frac{150}{60}$

                   $\frac{3\not630}{6\not60}=\frac{330}{60}$

                   $\frac{5\not610}{6\not60}=\frac{510}{60}$



denominator= 662


                   $\frac{1\not655}{6\not62}=\frac{155}{62}$

                   $\frac{3\not641}{6\not62}=\frac{341}{62}$

                   $\frac{5\not627}{6\not62}=\frac{527}{62}$

                   $\frac{7\not613}{6\not62}=\frac{713}{62}$



denominator= 664


                   $\frac{1\not660}{6\not64}=\frac{160}{64}$

                   $\frac{3\not652}{6\not64}=\frac{352}{64}$

                   $\frac{5\not644}{6\not64}=\frac{544}{64}$

                   $\frac{7\not636}{6\not64}=\frac{736}{64}$

                   $\frac{9\not628}{6\not64}=\frac{928}{64}$

                  $\frac{11\not620}{6\not64}=\frac{1120}{64}$

                  $\frac{13\not612}{6\not64}=\frac{1312}{64}$

                  $\frac{15\not604}{6\not64}=\frac{1504}{64}$



denominator= 671


                    $\frac{\not770}{6\not71}=\frac{70}{61}$

                   $\frac{1\not760}{6\not71}=\frac{160}{61}$

                   $\frac{2\not750}{6\not71}=\frac{250}{61}$

                   $\frac{3\not740}{6\not71}=\frac{340}{61}$

                   $\frac{4\not730}{6\not71}=\frac{430}{61}$

                   $\frac{5\not720}{6\not71}=\frac{520}{61}$

                   $\frac{7\not700}{6\not71}=\frac{700}{61}$



denominator= 682


                    $\frac{\not880}{6\not82}=\frac{80}{62}$

                   $\frac{1\not870}{6\not82}=\frac{170}{62}$

                   $\frac{2\not860}{6\not82}=\frac{260}{62}$

                   $\frac{3\not850}{6\not82}=\frac{350}{62}$

                   $\frac{4\not840}{6\not82}=\frac{440}{62}$

                   $\frac{5\not830}{6\not82}=\frac{530}{62}$

                   $\frac{7\not810}{6\not82}=\frac{710}{62}$

                   $\frac{8\not800}{6\not82}=\frac{800}{62}$



denominator= 693


                    $\frac{\not990}{6\not93}=\frac{90}{63}$

                   $\frac{1\not980}{6\not93}=\frac{180}{63}$

                   $\frac{2\not970}{6\not93}=\frac{270}{63}$

                   $\frac{3\not960}{6\not93}=\frac{360}{63}$

                   $\frac{4\not950}{6\not93}=\frac{450}{63}$

                   $\frac{5\not940}{6\not93}=\frac{540}{63}$

                   $\frac{7\not920}{6\not93}=\frac{720}{63}$

                   $\frac{8\not910}{6\not93}=\frac{810}{63}$

                   $\frac{9\not900}{6\not93}=\frac{900}{63}$



denominator= 754


                  $\frac{24\not505}{7\not54}=\frac{2405}{74}$



denominator= 770


                   $\frac{1\not760}{7\not70}=\frac{160}{70}$

                   $\frac{2\not750}{7\not70}=\frac{250}{70}$

                   $\frac{3\not740}{7\not70}=\frac{340}{70}$

                   $\frac{4\not730}{7\not70}=\frac{430}{70}$

                   $\frac{5\not720}{7\not70}=\frac{520}{70}$

                   $\frac{6\not710}{7\not70}=\frac{610}{70}$



denominator= 781


                    $\frac{\not880}{7\not81}=\frac{80}{71}$

                   $\frac{1\not870}{7\not81}=\frac{170}{71}$

                   $\frac{2\not860}{7\not81}=\frac{260}{71}$

                   $\frac{3\not850}{7\not81}=\frac{350}{71}$

                   $\frac{4\not840}{7\not81}=\frac{440}{71}$

                   $\frac{5\not830}{7\not81}=\frac{530}{71}$

                   $\frac{6\not820}{7\not81}=\frac{620}{71}$

                   $\frac{8\not800}{7\not81}=\frac{800}{71}$



denominator= 790


                   $\frac{1\not975}{7\not90}=\frac{175}{70}$

                   $\frac{5\not925}{7\not90}=\frac{525}{70}$



denominator= 792


                    $\frac{\not990}{7\not92}=\frac{90}{72}$

                   $\frac{1\not980}{7\not92}=\frac{180}{72}$

                   $\frac{2\not970}{7\not92}=\frac{270}{72}$

                   $\frac{4\not950}{7\not92}=\frac{450}{72}$

                   $\frac{5\not940}{7\not92}=\frac{540}{72}$

                   $\frac{6\not930}{7\not92}=\frac{630}{72}$

                   $\frac{8\not910}{7\not92}=\frac{810}{72}$

                   $\frac{9\not900}{7\not92}=\frac{900}{72}$



denominator= 794


                   $\frac{1\not985}{7\not94}=\frac{185}{74}$

                   $\frac{5\not955}{7\not94}=\frac{555}{74}$

                   $\frac{9\not925}{7\not94}=\frac{925}{74}$



denominator= 796


                    $\frac{\not995}{7\not96}=\frac{95}{76}$

                   $\frac{1\not990}{7\not96}=\frac{190}{76}$

                   $\frac{2\not985}{7\not96}=\frac{285}{76}$

                   $\frac{4\not975}{7\not96}=\frac{475}{76}$

                   $\frac{5\not970}{7\not96}=\frac{570}{76}$

                   $\frac{6\not965}{7\not96}=\frac{665}{76}$

                   $\frac{8\not955}{7\not96}=\frac{855}{76}$

                   $\frac{9\not950}{7\not96}=\frac{950}{76}$

                  $\frac{10\not945}{7\not96}=\frac{1045}{76}$

                  $\frac{12\not935}{7\not96}=\frac{1235}{76}$

                  $\frac{13\not930}{7\not96}=\frac{1330}{76}$

                  $\frac{14\not925}{7\not96}=\frac{1425}{76}$

                  $\frac{16\not915}{7\not96}=\frac{1615}{76}$

                  $\frac{17\not910}{7\not96}=\frac{1710}{76}$

                  $\frac{18\not905}{7\not96}=\frac{1805}{76}$



denominator= 798


                   $\frac{1\not995}{7\not98}=\frac{195}{78}$

                   $\frac{5\not985}{7\not98}=\frac{585}{78}$

                   $\frac{9\not975}{7\not98}=\frac{975}{78}$

                  $\frac{13\not965}{7\not98}=\frac{1365}{78}$

                  $\frac{17\not955}{7\not98}=\frac{1755}{78}$

                  $\frac{21\not945}{7\not98}=\frac{2145}{78}$

                  $\frac{25\not935}{7\not98}=\frac{2535}{78}$

                  $\frac{29\not925}{7\not98}=\frac{2925}{78}$

                  $\frac{33\not915}{7\not98}=\frac{3315}{78}$

                  $\frac{37\not905}{7\not98}=\frac{3705}{78}$



denominator= 830


                    $\frac{\not332}{8\not30}=\frac{32}{80}$

                   $\frac{1\not328}{8\not30}=\frac{128}{80}$

                   $\frac{2\not324}{8\not30}=\frac{224}{80}$

                   $\frac{4\not316}{8\not30}=\frac{416}{80}$

                   $\frac{5\not312}{8\not30}=\frac{512}{80}$

                   $\frac{6\not308}{8\not30}=\frac{608}{80}$

                   $\frac{7\not304}{8\not30}=\frac{704}{80}$



denominator= 844


                  $\frac{27\not430}{8\not44}=\frac{2730}{84}$

                  $\frac{65\not410}{8\not44}=\frac{6510}{84}$



denominator= 854


                  $\frac{19\not520}{8\not54}=\frac{1920}{84}$

                  $\frac{30\not500}{8\not54}=\frac{3000}{84}$



denominator= 880


                   $\frac{1\not870}{8\not80}=\frac{170}{80}$

                   $\frac{2\not860}{8\not80}=\frac{260}{80}$

                   $\frac{3\not850}{8\not80}=\frac{350}{80}$

                   $\frac{4\not840}{8\not80}=\frac{440}{80}$

                   $\frac{5\not830}{8\not80}=\frac{530}{80}$

                   $\frac{6\not820}{8\not80}=\frac{620}{80}$

                   $\frac{7\not810}{8\not80}=\frac{710}{80}$



denominator= 882


                   $\frac{4\not851}{8\not82}=\frac{451}{82}$



denominator= 884


                   $\frac{2\not873}{8\not84}=\frac{273}{84}$

                   $\frac{4\not862}{8\not84}=\frac{462}{84}$

                   $\frac{6\not851}{8\not84}=\frac{651}{84}$

                  $\frac{10\not829}{8\not84}=\frac{1029}{84}$

                  $\frac{12\not818}{8\not84}=\frac{1218}{84}$

                  $\frac{14\not807}{8\not84}=\frac{1407}{84}$



denominator= 886


                   $\frac{4\not873}{8\not86}=\frac{473}{86}$

                  $\frac{12\not847}{8\not86}=\frac{1247}{86}$

                  $\frac{20\not821}{8\not86}=\frac{2021}{86}$



denominator= 891


                    $\frac{\not990}{8\not91}=\frac{90}{81}$

                   $\frac{1\not980}{8\not91}=\frac{180}{81}$

                   $\frac{2\not970}{8\not91}=\frac{270}{81}$

                   $\frac{3\not960}{8\not91}=\frac{360}{81}$

                   $\frac{4\not950}{8\not91}=\frac{450}{81}$

                   $\frac{5\not940}{8\not91}=\frac{540}{81}$

                   $\frac{6\not930}{8\not91}=\frac{630}{81}$

                   $\frac{7\not920}{8\not91}=\frac{720}{81}$

                   $\frac{9\not900}{8\not91}=\frac{900}{81}$



denominator= 894


                   $\frac{2\not980}{8\not94}=\frac{280}{84}$

                   $\frac{5\not960}{8\not94}=\frac{560}{84}$

                  $\frac{11\not920}{8\not94}=\frac{1120}{84}$

                  $\frac{14\not900}{8\not94}=\frac{1400}{84}$



denominator= 897


                   $\frac{2\not990}{8\not97}=\frac{290}{87}$

                   $\frac{5\not980}{8\not97}=\frac{580}{87}$

                  $\frac{11\not960}{8\not97}=\frac{1160}{87}$

                  $\frac{14\not950}{8\not97}=\frac{1450}{87}$

                  $\frac{20\not930}{8\not97}=\frac{2030}{87}$

                  $\frac{23\not920}{8\not97}=\frac{2320}{87}$

                  $\frac{29\not900}{8\not97}=\frac{2900}{87}$



denominator= 930


                   $\frac{2\not325}{9\not30}=\frac{225}{90}$



denominator= 931


                   $\frac{1\not330}{9\not31}=\frac{130}{91}$

                   $\frac{3\not325}{9\not31}=\frac{325}{91}$

                   $\frac{5\not320}{9\not31}=\frac{520}{91}$

                   $\frac{7\not315}{9\not31}=\frac{715}{91}$

                  $\frac{11\not305}{9\not31}=\frac{1105}{91}$

                  $\frac{13\not300}{9\not31}=\frac{1300}{91}$



denominator= 932


                   $\frac{2\not330}{9\not32}=\frac{230}{92}$

                  $\frac{16\not310}{9\not32}=\frac{1610}{92}$



denominator= 976


                  $\frac{20\not740}{9\not76}=\frac{2040}{96}$

                  $\frac{31\not720}{9\not76}=\frac{3120}{96}$

                  $\frac{42\not700}{9\not76}=\frac{4200}{96}$



   -------------------------------------------------------------------------



Section 5

All the Anomalous Cancellations with the denominator with, 3,

    digits, and where the , 1, -th digit of the numerator

          is cancelled out with the, 2,  -th digit of the denominator

denominator= 120


                     $\frac{\not12}{\not120}=\frac{2}{20}$



denominator= 130


                     $\frac{\not13}{\not130}=\frac{3}{30}$



denominator= 140


                     $\frac{\not14}{\not140}=\frac{4}{40}$



denominator= 150


                     $\frac{\not15}{\not150}=\frac{5}{50}$



denominator= 160


                     $\frac{\not16}{\not160}=\frac{6}{60}$



denominator= 170


                     $\frac{\not17}{\not170}=\frac{7}{70}$



denominator= 180


                     $\frac{\not18}{\not180}=\frac{8}{80}$



denominator= 190


                     $\frac{\not19}{\not190}=\frac{9}{90}$



denominator= 210


                     $\frac{\not21}{\not210}=\frac{1}{10}$



denominator= 222


                   $\frac{12\not21}{\not222}=\frac{121}{22}$



denominator= 240


                     $\frac{\not24}{\not240}=\frac{4}{40}$



denominator= 260


                     $\frac{\not26}{\not260}=\frac{6}{60}$



denominator= 280


                     $\frac{\not28}{\not280}=\frac{8}{80}$



denominator= 310


                     $\frac{\not31}{\not310}=\frac{1}{10}$



denominator= 320


                     $\frac{\not32}{\not320}=\frac{2}{20}$



denominator= 325


                    $\frac{1\not30}{\not325}=\frac{10}{25}$



denominator= 332


                    $\frac{8\not30}{\not332}=\frac{80}{32}$



denominator= 360


                     $\frac{\not36}{\not360}=\frac{6}{60}$



denominator= 370


                     $\frac{\not37}{\not370}=\frac{7}{70}$



denominator= 380


                     $\frac{\not38}{\not380}=\frac{8}{80}$



denominator= 390


                     $\frac{\not39}{\not390}=\frac{9}{90}$



denominator= 410


                     $\frac{\not41}{\not410}=\frac{1}{10}$



denominator= 420


                     $\frac{\not42}{\not420}=\frac{2}{20}$



denominator= 430


                     $\frac{\not43}{\not430}=\frac{3}{30}$



denominator= 444


                   $\frac{14\not43}{\not444}=\frac{143}{44}$

                   $\frac{24\not42}{\not444}=\frac{242}{44}$

                   $\frac{34\not41}{\not444}=\frac{341}{44}$



denominator= 510


                     $\frac{\not51}{\not510}=\frac{1}{10}$



denominator= 520


                     $\frac{\not52}{\not520}=\frac{2}{20}$



denominator= 530


                     $\frac{\not53}{\not530}=\frac{3}{30}$



denominator= 540


                     $\frac{\not54}{\not540}=\frac{4}{40}$



denominator= 555


                   $\frac{15\not54}{\not555}=\frac{154}{55}$

                   $\frac{25\not53}{\not555}=\frac{253}{55}$

                   $\frac{35\not52}{\not555}=\frac{352}{55}$

                   $\frac{45\not51}{\not555}=\frac{451}{55}$



denominator= 610


                     $\frac{\not61}{\not610}=\frac{1}{10}$



denominator= 620


                     $\frac{\not62}{\not620}=\frac{2}{20}$



denominator= 630


                     $\frac{\not63}{\not630}=\frac{3}{30}$



denominator= 640


                     $\frac{\not64}{\not640}=\frac{4}{40}$

                    $\frac{1\not60}{\not640}=\frac{10}{40}$



denominator= 644


                    $\frac{1\not61}{\not644}=\frac{11}{44}$



denominator= 648


                    $\frac{1\not62}{\not648}=\frac{12}{48}$



denominator= 650


                     $\frac{\not65}{\not650}=\frac{5}{50}$

                    $\frac{2\not60}{\not650}=\frac{20}{50}$



denominator= 652


                    $\frac{1\not63}{\not652}=\frac{13}{52}$



denominator= 655


                    $\frac{2\not62}{\not655}=\frac{22}{55}$



denominator= 656


                    $\frac{1\not64}{\not656}=\frac{14}{56}$



denominator= 664


                    $\frac{1\not66}{\not664}=\frac{16}{64}$

                   $\frac{11\not62}{\not664}=\frac{112}{64}$

                   $\frac{16\not60}{\not664}=\frac{160}{64}$



denominator= 665


                    $\frac{2\not66}{\not665}=\frac{26}{65}$

                   $\frac{10\not64}{\not665}=\frac{104}{65}$

                   $\frac{14\not63}{\not665}=\frac{143}{65}$

                   $\frac{18\not62}{\not665}=\frac{182}{65}$

                   $\frac{22\not61}{\not665}=\frac{221}{65}$



denominator= 666


                   $\frac{16\not65}{\not666}=\frac{165}{66}$

                   $\frac{36\not63}{\not666}=\frac{363}{66}$

                   $\frac{56\not61}{\not666}=\frac{561}{66}$



denominator= 710


                     $\frac{\not71}{\not710}=\frac{1}{10}$



denominator= 720


                     $\frac{\not72}{\not720}=\frac{2}{20}$



denominator= 730


                     $\frac{\not73}{\not730}=\frac{3}{30}$



denominator= 740


                     $\frac{\not74}{\not740}=\frac{4}{40}$



denominator= 750


                     $\frac{\not75}{\not750}=\frac{5}{50}$



denominator= 756


                    $\frac{2\not70}{\not756}=\frac{20}{56}$



denominator= 760


                     $\frac{\not76}{\not760}=\frac{6}{60}$



denominator= 775


                   $\frac{21\not70}{\not775}=\frac{210}{75}$



denominator= 777


                   $\frac{17\not76}{\not777}=\frac{176}{77}$

                   $\frac{27\not75}{\not777}=\frac{275}{77}$

                   $\frac{37\not74}{\not777}=\frac{374}{77}$

                   $\frac{47\not73}{\not777}=\frac{473}{77}$

                   $\frac{57\not72}{\not777}=\frac{572}{77}$

                   $\frac{67\not71}{\not777}=\frac{671}{77}$



denominator= 810


                     $\frac{\not81}{\not810}=\frac{1}{10}$



denominator= 820


                     $\frac{\not82}{\not820}=\frac{2}{20}$



denominator= 830


                     $\frac{\not83}{\not830}=\frac{3}{30}$



denominator= 840


                     $\frac{\not84}{\not840}=\frac{4}{40}$



denominator= 850


                     $\frac{\not85}{\not850}=\frac{5}{50}$



denominator= 860


                     $\frac{\not86}{\not860}=\frac{6}{60}$



denominator= 870


                     $\frac{\not87}{\not870}=\frac{7}{70}$



denominator= 888


                   $\frac{18\not87}{\not888}=\frac{187}{88}$

                   $\frac{28\not86}{\not888}=\frac{286}{88}$

                   $\frac{38\not85}{\not888}=\frac{385}{88}$

                   $\frac{48\not84}{\not888}=\frac{484}{88}$

                   $\frac{58\not83}{\not888}=\frac{583}{88}$

                   $\frac{68\not82}{\not888}=\frac{682}{88}$

                   $\frac{78\not81}{\not888}=\frac{781}{88}$



denominator= 910


                     $\frac{\not91}{\not910}=\frac{1}{10}$



denominator= 920


                     $\frac{\not92}{\not920}=\frac{2}{20}$



denominator= 930


                     $\frac{\not93}{\not930}=\frac{3}{30}$



denominator= 940


                     $\frac{\not94}{\not940}=\frac{4}{40}$



denominator= 950


                     $\frac{\not95}{\not950}=\frac{5}{50}$

                    $\frac{1\not90}{\not950}=\frac{10}{50}$



denominator= 955


                    $\frac{1\not91}{\not955}=\frac{11}{55}$



denominator= 960


                     $\frac{\not96}{\not960}=\frac{6}{60}$

                    $\frac{1\not92}{\not960}=\frac{12}{60}$



denominator= 965


                    $\frac{1\not93}{\not965}=\frac{13}{65}$



denominator= 970


                     $\frac{\not97}{\not970}=\frac{7}{70}$

                    $\frac{1\not94}{\not970}=\frac{14}{70}$

                    $\frac{2\not91}{\not970}=\frac{21}{70}$



denominator= 975


                    $\frac{1\not95}{\not975}=\frac{15}{75}$

                    $\frac{3\not90}{\not975}=\frac{30}{75}$



denominator= 980


                     $\frac{\not98}{\not980}=\frac{8}{80}$

                    $\frac{1\not96}{\not980}=\frac{16}{80}$

                    $\frac{2\not94}{\not980}=\frac{24}{80}$

                    $\frac{3\not92}{\not980}=\frac{32}{80}$

                    $\frac{4\not90}{\not980}=\frac{40}{80}$



denominator= 982


                    $\frac{4\not91}{\not982}=\frac{41}{82}$



denominator= 984


                    $\frac{4\not92}{\not984}=\frac{42}{84}$



denominator= 985


                    $\frac{1\not97}{\not985}=\frac{17}{85}$

                    $\frac{3\not94}{\not985}=\frac{34}{85}$

                    $\frac{5\not91}{\not985}=\frac{51}{85}$



denominator= 986


                    $\frac{4\not93}{\not986}=\frac{43}{86}$



denominator= 988


                    $\frac{4\not94}{\not988}=\frac{44}{88}$



denominator= 992


                    $\frac{4\not96}{\not992}=\frac{46}{92}$



denominator= 994


                    $\frac{4\not97}{\not994}=\frac{47}{94}$

                   $\frac{14\not91}{\not994}=\frac{141}{94}$



denominator= 995


                    $\frac{1\not99}{\not995}=\frac{19}{95}$

                    $\frac{3\not98}{\not995}=\frac{38}{95}$

                    $\frac{5\not97}{\not995}=\frac{57}{95}$

                    $\frac{7\not96}{\not995}=\frac{76}{95}$

                   $\frac{11\not94}{\not995}=\frac{114}{95}$

                   $\frac{13\not93}{\not995}=\frac{133}{95}$

                   $\frac{15\not92}{\not995}=\frac{152}{95}$

                   $\frac{17\not91}{\not995}=\frac{171}{95}$



denominator= 996


                    $\frac{4\not98}{\not996}=\frac{48}{96}$

                   $\frac{14\not94}{\not996}=\frac{144}{96}$

                   $\frac{24\not90}{\not996}=\frac{240}{96}$



denominator= 998


                    $\frac{4\not99}{\not998}=\frac{49}{98}$

                   $\frac{14\not97}{\not998}=\frac{147}{98}$

                   $\frac{24\not95}{\not998}=\frac{245}{98}$

                   $\frac{34\not93}{\not998}=\frac{343}{98}$

                   $\frac{44\not91}{\not998}=\frac{441}{98}$



   -------------------------------------------------------------------------



Section 6

All the Anomalous Cancellations with the denominator with, 3,

    digits, and where the , 2, -th digit of the numerator

          is cancelled out with the, 2,  -th digit of the denominator

denominator= 220


                   $\frac{1\not210}{\not220}=\frac{110}{20}$



denominator= 432


                   $\frac{1\not404}{\not432}=\frac{104}{32}$



denominator= 436


                   $\frac{1\not417}{\not436}=\frac{117}{36}$



denominator= 438


                   $\frac{2\not409}{\not438}=\frac{209}{38}$



denominator= 440


                   $\frac{1\not430}{\not440}=\frac{130}{40}$

                   $\frac{2\not420}{\not440}=\frac{220}{40}$

                   $\frac{3\not410}{\not440}=\frac{310}{40}$



denominator= 442


                   $\frac{2\not431}{\not442}=\frac{231}{42}$

                   $\frac{6\not409}{\not442}=\frac{609}{42}$



denominator= 540


                   $\frac{1\not512}{\not540}=\frac{112}{40}$



denominator= 545


                   $\frac{1\not526}{\not545}=\frac{126}{45}$

                   $\frac{2\not507}{\not545}=\frac{207}{45}$



denominator= 550


                   $\frac{1\not540}{\not550}=\frac{140}{50}$

                   $\frac{2\not530}{\not550}=\frac{230}{50}$

                   $\frac{3\not520}{\not550}=\frac{320}{50}$

                   $\frac{4\not510}{\not550}=\frac{410}{50}$



denominator= 640


                   $\frac{1\not600}{\not640}=\frac{100}{40}$



denominator= 642


                   $\frac{1\not605}{\not642}=\frac{105}{42}$



denominator= 644


                   $\frac{1\not610}{\not644}=\frac{110}{44}$



denominator= 646


                   $\frac{1\not615}{\not646}=\frac{115}{46}$



denominator= 648


                   $\frac{1\not620}{\not648}=\frac{120}{48}$



denominator= 650


                   $\frac{1\not625}{\not650}=\frac{125}{50}$



denominator= 652


                   $\frac{1\not630}{\not652}=\frac{130}{52}$



denominator= 654


                   $\frac{1\not635}{\not654}=\frac{135}{54}$



denominator= 656


                   $\frac{1\not640}{\not656}=\frac{140}{56}$

                   $\frac{3\not608}{\not656}=\frac{308}{56}$



denominator= 658


                   $\frac{1\not645}{\not658}=\frac{145}{58}$

                   $\frac{3\not619}{\not658}=\frac{319}{58}$



denominator= 660


                   $\frac{1\not650}{\not660}=\frac{150}{60}$

                   $\frac{3\not630}{\not660}=\frac{330}{60}$

                   $\frac{5\not610}{\not660}=\frac{510}{60}$



denominator= 662


                   $\frac{1\not655}{\not662}=\frac{155}{62}$

                   $\frac{3\not641}{\not662}=\frac{341}{62}$

                   $\frac{5\not627}{\not662}=\frac{527}{62}$

                   $\frac{7\not613}{\not662}=\frac{713}{62}$



denominator= 664


                   $\frac{1\not660}{\not664}=\frac{160}{64}$

                   $\frac{3\not652}{\not664}=\frac{352}{64}$

                   $\frac{5\not644}{\not664}=\frac{544}{64}$

                   $\frac{7\not636}{\not664}=\frac{736}{64}$

                   $\frac{9\not628}{\not664}=\frac{928}{64}$

                  $\frac{11\not620}{\not664}=\frac{1120}{64}$

                  $\frac{13\not612}{\not664}=\frac{1312}{64}$

                  $\frac{15\not604}{\not664}=\frac{1504}{64}$



denominator= 749


                   $\frac{1\not712}{\not749}=\frac{112}{49}$



denominator= 756


                   $\frac{1\not728}{\not756}=\frac{128}{56}$

                   $\frac{2\not700}{\not756}=\frac{200}{56}$



denominator= 763


                   $\frac{1\not744}{\not763}=\frac{144}{63}$

                   $\frac{2\not725}{\not763}=\frac{225}{63}$

                   $\frac{3\not706}{\not763}=\frac{306}{63}$



denominator= 770


                   $\frac{1\not760}{\not770}=\frac{160}{70}$

                   $\frac{2\not750}{\not770}=\frac{250}{70}$

                   $\frac{3\not740}{\not770}=\frac{340}{70}$

                   $\frac{4\not730}{\not770}=\frac{430}{70}$

                   $\frac{5\not720}{\not770}=\frac{520}{70}$

                   $\frac{6\not710}{\not770}=\frac{610}{70}$



denominator= 848


                   $\frac{1\not802}{\not848}=\frac{102}{48}$



denominator= 856


                   $\frac{1\not819}{\not856}=\frac{119}{56}$



denominator= 864


                   $\frac{1\not836}{\not864}=\frac{136}{64}$

                   $\frac{2\not808}{\not864}=\frac{208}{64}$



denominator= 868


                   $\frac{2\not821}{\not868}=\frac{221}{68}$



denominator= 872


                   $\frac{1\not853}{\not872}=\frac{153}{72}$

                   $\frac{2\not834}{\not872}=\frac{234}{72}$

                   $\frac{3\not815}{\not872}=\frac{315}{72}$



denominator= 874


                   $\frac{4\not807}{\not874}=\frac{407}{74}$



denominator= 876


                   $\frac{2\not847}{\not876}=\frac{247}{76}$

                   $\frac{4\not818}{\not876}=\frac{418}{76}$



denominator= 878


                   $\frac{4\not829}{\not878}=\frac{429}{78}$



denominator= 880


                   $\frac{1\not870}{\not880}=\frac{170}{80}$

                   $\frac{2\not860}{\not880}=\frac{260}{80}$

                   $\frac{3\not850}{\not880}=\frac{350}{80}$

                   $\frac{4\not840}{\not880}=\frac{440}{80}$

                   $\frac{5\not830}{\not880}=\frac{530}{80}$

                   $\frac{6\not820}{\not880}=\frac{620}{80}$

                   $\frac{7\not810}{\not880}=\frac{710}{80}$



denominator= 882


                   $\frac{4\not851}{\not882}=\frac{451}{82}$



denominator= 884


                   $\frac{2\not873}{\not884}=\frac{273}{84}$

                   $\frac{4\not862}{\not884}=\frac{462}{84}$

                   $\frac{6\not851}{\not884}=\frac{651}{84}$

                  $\frac{10\not829}{\not884}=\frac{1029}{84}$

                  $\frac{12\not818}{\not884}=\frac{1218}{84}$

                  $\frac{14\not807}{\not884}=\frac{1407}{84}$



denominator= 886


                   $\frac{4\not873}{\not886}=\frac{473}{86}$

                  $\frac{12\not847}{\not886}=\frac{1247}{86}$

                  $\frac{20\not821}{\not886}=\frac{2021}{86}$



   -------------------------------------------------------------------------

          This ends this book that took, 2.980, seconds to generate.



\bigskip
{\bf References}

\bigskip
\hrule
\bigskip
Shalosh B. Ekhad, c/o D. Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt ShaloshBEkhad at gmail dot com}   \quad .


\end

