Use the euclidean algorithm to prove that if gcd(a,b) = 1 and a|c and b|c then ab|c

Question

I am a bit confused here.

I assume that: $gcd(a, b) = 1 \wedge c = ax_{1} \wedge c = bx_{2}$.

I tried to find a formula starting from $a = bx_{3} + r$. But I didn't succeed, any tips?

Answer 1

Using the Euclidean algorithm

$$1=ax+by$$

Then as $c=ak$ and $c=bj$ we can multiply our main equation by $c$ on both sides (and replace it on the right by $bj$ and $ak$) to get

$$c=(ab)jx+(ab)ky$$

so $c=ab(jx+ky)$.