Is the statement "$(m, n)=d$ if and only if there exist integers $r$ and $s$ such that $r m+s n=d$" problematic?

Question

My textbook Groups, Matrices, and Vector Spaces - A Group Theoretic Approach to Linear Algebra by James B. Carrell said that

$(m, n)=d$ if and only if there exist integers $r$ and $s$ such that $r m+s n=d$.

We have $2\cdot3+4\cdot5=26$, but $(3,5)=1\neq 26$. Could you please explain if this paragraph is problematic?

Answer 1

The first implication says there exists such two integers, not saying that any two integers you pick will fit the bill. For the case of (3,5) we can find 7*3+(-4)*5=1. So there exists integers. Now I notice the if and only if, so the reverse implication also must hold. It looks problematic like this. You are right. By this definition if d is gcd, than any multiple of d will also be gcd, which is wrong. The correct statement will be d, the smallest positive number for which there exists such two r and s will be gcd.

In other words d will be gcd if it is a positive generator for the ideal generated by m and n.

Answer 2

While true for $d=26$, the 'if and only if' bit means that if we can find a solution to $rm+sn=d$ for some positive integer $d$, and this $d$ is smaller than all the others, then this is the value of $(m,n)$, and in the case $(3,5)$ we have $2\cdot3+(-1)\cdot5=1$.