If , then .
My knowledge of the greatest common factor (GCD) is rather limited, but I'm thinking this will involve representing GCD expressions with linear combinations. Here is what I know:
The greatest common divisor of the nonzero integers and is the least positive integer that is a linear combination of and .
And the trivial assumption: implies there exists a number that is a common divisor of and .
Then divides , , and .
So , , and for some integers , , and .
Then divides and .
So and for some integers and .
If this is even the correct start to this proof, I would be amazed. I imagine the next steps would involve substitution, but I'm not sure how to proceed. Any help would be appreciated!