%q2S.tex: 
%begin macros
\baselineskip=14pt
\parskip=10pt
\magnification=\magstephalf
\parindent=0pt
\overfullrule=0in
\def\frac#1#2{{#1 \over #2}}
\def\g{\bigtriangledown}
\def\S{\cal S}
\nopagenumbers
%end macros
 

\centerline
{
\bf  Solutions to Quiz \# 2 for Dr. Z.'s Number Theory
Course for Oct. 3, 2013
}
\bigskip
{\bf 1.} (5 points)
Use any method to find the  proudct-of-prime-powers representation of

$$
(3^3 \cdot 5^{98} \cdot 11^{14} \cdot 43) \cdot (3^2 \cdot 5^{3}  \cdot 11^{12} \cdot 13^{2} \cdot 43^2) \quad .
$$

{\bf Sol. of 1}:


$$
(3^3 \cdot 5^{98} \cdot 11^{14} \cdot 43) \cdot (3^2 \cdot 5^{3}  \cdot 11^{12} \cdot 13^{2} \cdot 43^2) 
$$
$$
=(3^3 \cdot 5^{98} \cdot 11^{14} \cdot 13^{0} \cdot 43^1) \cdot (3^2 \cdot 5^{3}  \cdot 11^{12} \cdot 13^{2} \cdot 43^2) 
$$
$$
=3^{3+2} \cdot 5^{98+3} \cdot 11^{14+12} \cdot 13^{0+2} \cdot 43^{1+2}
$$
$$
=3^{5} \cdot 5^{101} \cdot 11^{26} \cdot 13^{2} \cdot 43^{3} \quad .
$$
{\bf Ans. to 1}: $3^{5} \cdot 5^{101} \cdot 11^{26} \cdot 13^{2} \cdot 43^{3}$ .

{\bf 2.} (5 points)
Use the clever way to find  $Div(7007)$.

{\bf Sol. of 2}:
$$
7007= 7^2 \cdot 11 \cdot 13 \quad
$$
So
$$
Div(7007)=\{1,7,7^2 \} \cdot \{1,11 \} \cdot \{ 1, 13 \} 
$$
$$
\{1,7,7^2 \} \cdot (\{1,11 \} \cdot \{ 1, 13 \} )=
\{1,7,49 \} \cdot (\{1,11, 13, 143 \}
$$
$$
=\{1,11, 13, 143 ,
$$
$$
7,77, 91, 1001,
$$
$$
49, 539, 637, 7007 \}
$$
$$
\{1, 7, 11, 13, 49, 77, 91, 143, 539, 637, 1001, 7007 \} \quad .
$$

{\bf Ans. to 2}: $Div(7007)=\{1, 7, 11, 13, 49, 77, 91, 143, 539, 637, 1001, 7007 \}$.

\end
