I was comparing the statement of the ABC conjecture on Wikipedia to an article I found geared towards laymen. The propositions differ slightly, but they would be equivalent if the following were true:
If $a$ and $b$ are coprime, and $a + b = c$, is $c$ coprime to both $a$ and $b$?
By division algorithm, if (without loss of generality) $a > b$, then $a$ and $c$ must be coprime: the first iteration of the algorithm subtracts $c = 1*a + b$, and then you're left with $a$ and $b$, which are coprime by proposition and so generate 1 as the final result.
If $b$ is prime, I can imagine a proof by contradiction via the division algorithm.
For more complicated $b$ I am not sure it holds.