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## Homework Statement

(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.

## Homework Equations

## 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

x

_{0}a+y

_{0}b=d

Then we multiply both sides by some arbitrary element, such as n. Giving us,

(nx

_{0})a+(ny

_{0})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.