Hello could anyone help me prove this Let b g k and m be int

Hello, could anyone help me prove this:

Let b, g, k and m be integers with b = 3k + 1. Prove if gcd(b, m) = g, then gcd(3, g) = 1.

Aside from stating the assumtions I don\'t have a clue what to do next.

Solution

g is gcd of b then b is divisble by g, therefore

If a = bq + r, then GCD(a, b) = GCD(b, r).

Proof. We will show that if a = bq + r, then an integer d is a common divisor of a and b if, and only if, d is a common divisor of b and r. Let d be a common divisor of a and b. Then d|a and d|b. Thus d|(a bq), which means d|r, since r = a bq. Thus d is a common divisor of b and r. Now suppose d is a common divisor of b and r. Then d|b and d|r. Thus d|(bq +r), so d|a. Therefore, d must be a common divisor of a and b. Thus, the set of common divisors of a and b are the same as the set of common divisors of b and r. It follows that d is the greatest common divisor of a and b if and only if d is the greatest common divisor of b and r.