example, take two numbers a and b where you know the gcd is 1. so we have ma + nb = 1 for some integers m and n. now multiply that equation by 2, we get
(2m)a + (2n)b = 2, write 2m as x and 2n as y, we have xa + yb = 2
but we know the gcd is 1! what can you say therefore?
yes, if gcd(a,b) = 2, then you can write a linear combination of a and b equaling 2So I know you can do it the other way around, so gcd(a,b)=2 does equal ax+by=2 - but I wasn't sure you could switch it. Even then, I would need more proof that it doesn't work. Some kind of guidance would be lovely.