I would like some help on the problem above, it is a part of a school problem set but I'm having a bit of trouble with the explanation for it. The question is as follows:
Suppose x and y are relatively prime integers. What are all possible values for gcd(7x−y, x+ 2y)? Explain.
From what I understand, the only common prime factor between x and y has to be 1 and from trial and error, I find that the resulting gcd also seems to 1. Could anyone please help me show how this could be written out and explained? Thank you!
Best Answer
Since the $\gcd$ has to be a divisor of $7x-y$ and $x+2y$, it has to divide twice the first plus the second, or $15x$. It also has to divide the $7$ times the second minus the first, or $15y$. Therefore, it has to divide $\gcd(15x,15y)=15$. Thus the four possible values are $1,3,5,15$ obtained on the pairs e.g $(2,1),(1,1),(1,2),(4,13)$.