Let a = 242, b = 142. Let g be the gretest common divisor of a and b, Find g, and find integers m,n such that g = m*a+n*n

I found g = 2

So 2 = 242m + n^2

how can i find m and n?

Also what is the fastest way to prove the following:

if a,b are integers, g is a natural number, and g is the GCD of a and b, then there exist integrs m,n suc that g = m*a+n*b?

Everywhere i check online they are giving me longgggg solutions,but what if this question was given on an exam that is an hour long, i had it on my midterm this semester