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{
\bf  Quiz \# 8 for Dr. Z.'s Number Theory
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{\bf NAME:} (print!) $\underline
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{\bf E-MAIL ADDRESS:} (print!) 
$\underline
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{\bf 1.} Find $\sigma(200)$

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

{\bf 2} (a) Define a perfect number

(b) Prove that if both $p$ and $2^p-1$ are primes, then
$$
2^{p-1} \cdot (2^p -1) \quad 
$$
is a perfect number.

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{\bf 1.} Find $\sigma(200)$

{\bf Sol. to 1}: First we factorize 200 
$$
200=2^3 \cdot 5^2 \quad
$$

Now
$$
\sigma(200)=(1+2+2^2+2^3) \cdot (1+5+5^2)=15 \cdot 31=465 \quad.
$$

{\bf Answer to 1}: $\sigma(200)=465$

{\bf 2} (a) Define a perfect number

{\bf Sol to 2}: A positive integer  is perfect if it equals to the sum of its proper divisors, or equivalenly
the sum of the divisors is twice that number. In symbols: $\sigma(2n)=2\sigma(n)$.


(b) Prove that if both $p$ and $2^p-1$ are primes, then
$$
2^{p-1} \cdot (2^p -1) \quad 
$$
is a perfect number.


