Use the Euclidean algorithm to show that 15m + 7 and 2m + 1 are relatively prime

Question

I am familiar with the Euclidean algorithm but I do not understand how to show that two numbers are relatively prime if there is an m

Answer 1

The key is the well known fact that if $a=qb+r$ where $0\le r\lt b$ then $(a,b)=(b,r)$ (equality of greatest common divisors). From which we have $$15m+7=7(2m+1)+m\Rightarrow (15m+7,2m+1)=(2m+1,m)$$ and we are done because $(2m+1,m)=1$.