1. The problem statement, all variables and given/known data (a) Let a and b be integers with gcd(a,b)=d, and assume that ma+nb=d for integers m and n. Show that the solutions in Z to xa+yb=d are exactly x=m+k(b/d), y=n-k(a/d) where k∈Z. (b) Let a and b be integers with gcd(a,b)=d. Show that the equation xa+yb=c has a solution (x,y)∈ ZxZ if and only if d|c. Z refers to integers. 2. Relevant equations 3. The attempt at a solution Ok so I'm pretty sure I figured out part (a). Part (b) on the other hand is causing me problems. I know since it's an "if and only if" question, I have to prove it both ways. This is what I have so far. So we have some x and y, such as x0a+y0b=d Then we multiply both sides by some arbitrary element, such as n. Giving us, (nx0)a+(ny0)b=nd=c So that's the first part. Now I'm supposed to go the other way with it, showing that xa+yb=c -> c=nd, n∈ Z.