If gcd(a, b) = 1 and a | bc. Prove that a | c.

Question

Let a, b, c ∈ N such that (a, b) = 1 and a | bc. Prove that a | c.

I'm a little confused about if I'm doing this proof right.

I know that $\exists p, q \in \mathbb{Z}$. Such that $pa = bc$ and $qa=c$. Re-arranging the first equation.

$$a = \frac{bc}{p} $$

Substituting this into the second equation.

$$ qa = c$$ $$ q(\frac{bc}{p}) = c$$ $$ \frac{qb}{p} c =c $$ $$ \frac{qb}{p} c =c $$

Thus this equation divides c. Therefore a | c. Does this proof make sense.

Thank you for any guidance.

Answer 1

You can't assume $c=qa$ because that if what you need to prove.

Hint for a solution: if $gcd(a,b)=1$ then there are numbers $k,l\in\mathbb{Z}$ such that $ak+bl=1$. Multiply this equation by $c$ and see what you get from there.

Answer 2

You do not know there is such a $q$. That is in fact exactly what you are trying to prove: it says $a$ divides $c$.

The standard argument for this theorem begins with the existence of integers $r$ and $s$ such that $$ ar + bs = 1 . $$

Multiply that equality by $c$ and try to conclude what you hope to prove.

Note: you should strive to write number theory proofs without fractions. Even when their use is correct it's confusing.