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\centerline
{
\bf  Solutions to Quiz \# 5 for Dr. Z.'s Number Theory
}
\bigskip


{\bf 1.} (5 points) Using divisiblity tests, determine
whether $517286$ is divisible by $11$.

{\bf Sol. of 1}: We take the {\it alternating sum}
$$
5-1+7-2+8-6 \,\, (mod \,\, 11)=4+5+2 \,\, (mod \,\, 11)=  11 \,\, (mod \,\, 11)= 0 \quad.
$$
So it is divisible by $11$.

{\bf 2.} (5 points) What is the day of the week of Halloween 2100.

{\bf Sol. of 2}: This year Oct. 31 is Thurs. so it is Day 5.

Oct. 31, 2100 will happen  in 2100-2024=76 years. There are $(2096-2024)/4=18$ leap days (Feb, 29) until then. Note that $2100$ itself is {\bf not} a leap year, since it is
a multiple of $100$ (and not a multlple of $400$).

So we have
$$
5+ 76 + 18 \,\, (mod \,\, 7) =5+6+4  \,\, (mod \,\, 7) =  1 \,\, (mod \,\, 7)  \quad.
$$

{\bf Ans. to 2}: Sunday.


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