1.Let a be an integer and p be a positive integer. Prove that if p divides a, then
GCD(a,p)=p.
2.Show that if GCD(a, c)=1 and c divides ab, then c divides b.
#1: Since $\displaystyle p \mid a$, we can say that $\displaystyle a = kp$ for some $\displaystyle k$.
So we're trying to find $\displaystyle d = (p, kp)$.
Since $\displaystyle d \mid p$ and $\displaystyle p$ is prime, $\displaystyle d = 1$ or $\displaystyle d = p$... Can you finish off?
#2:
$\displaystyle (a,c) =1 \ \Rightarrow \ ax + cy = 1 \ \iff \ abx + cby = b$. Can you conclude?
Huh.. It is not said that p is a prime integer..
__________________________________________________
Let d=gcd(a,p). We can say, in particular, that d divides p.
Since p divides both p (obvious) and a (because a is a multiple of p), we can say that p divides the gcd of a and p, that is d.
d | p and p | d
Hence p=d.
Note that the first $\displaystyle \Rightarrow$ can be an equivalence#2:
$\displaystyle (a,c) =1 \ \Rightarrow \ ax + cy = 1 \ \iff \ abx + cby = b$. Can you conclude?