I need help with number 4 Prove the following for any intege

Prove the following, for any integers a, b, and c. For each of these problems, you will need only the definition of the gcd. If a > 0 and a | b, then gcd(a, b) = a. gcd(a, 0) = a, if a > 0. gcd(a, b) = gcd(a, b + xa) for any x Z. Let p be a prime. Then gcd(a, p) = 1 or p. (Explain.)

Solution

Let a be an integer. Let p be a prime number.

Let d =gcd(a, p).

Then d|a and d|p.

If d|a then d must be 1 or a prime factor of a i.e p.

If d|p then d must be 1 or p

Thus d can be 1 or p.