- #1

scorpius1782

- 107

- 0

## Homework Statement

For integers a,b, and c, if a and c are relatively prime and c|ab, then c|b.

Knowing that: For any integers p and q, there are integers s and t such that gcd(p,q) = sp + tq.

The hint I'm given is that I should form an equation from the fact that they are "relatively prime."

The last caveat is that I cannot use fractions at all in my proof.

## Homework Equations

## The Attempt at a Solution

So my hang up is definitely the equation for relatively prime. As I understand it for relatively prime: gcd(a,c)=1. Then, I suppose I could just use the given equation to say that 1= sa + tc. (I believe this Bézout's identity).

Then I have the following claim: If 1= sa + tc and ab=cn then b=cp. Where n and p are just integers. Then I can say that sa=1-tc. Then multiplying the second condition by 's' I'd have sab=scn. Combining them b(1-tc)=scn.

But I can't do fractions so I'm not sure what to make of this, or if I've even started the problem correctly. I find it unlikely that my relatively prime equation is correct, or is what they intended me to do.

Thanks for the suggestions.