Chapter 10

Find the positive divisors of the following integers. a. 72 b. 31 c. 123

Prove that if $h/m$ and $m/n$, and $h/n$.

show that two integers divide each other if and only if they are equal.

Let $p$ and $q$ be two prime numbers. If $p=q+2$ then $p$ and $q$ are called "twin prime numbers." Find two pairs of twin prime numbers.

Prove that $\gcd (n,m)=\gcd (m,n)$.

Prove that if $m$ and $n$ are both even, then gcd $(m,n)=2\gcd (m/2,n/2)$.

Prove that if $n\ge m>0,$ then $\gcd (m,n)=\gcd$ $(m,n-m)$.