Linear Diophantine Equation ax + by = c?

Question

Prove that if gcd ( a ,b ) divides c , the eqaution has infinitly many solutions .Can you show this step by step . Thanks !

Answer 1

It would be easier to take an example
2x+3y=5
x=2-y +(1-y)/2 ( you make the division and write the remainder)
(1-y)/2 =t so y=1-2t
and x= 2-1+2t+t = 1+3t
Now giving t integer values you get infinite solutions
By Euclydes you must come to this type of expressions

Answer 2

I suggest you take a look at

http://mathworld.wolfram.com/DiophantineEquation.html

It's interesting.