A partial converse is, if there are integers A and B such that Am+Bn=1, then gcd(m,n)=1.

Answer correct but wrong question.

The following theorem is copied from Meserve, Fundamental Concepts of Algebra:

"Theorem 2-11. The greatest common divisor of any two positive integers m and n can be found as the last non-vanishing remainder n\(\displaystyle _{k}\) in the Euclidean Algorithm. There exist integers A and B such that

(2-10) (m,n) = n\(\displaystyle _{k}\) = Am + Bn"

For example: m=22, n=24

22=1x16+6, (22,16)=(16,6)

16=2x6+4, (16,6)=(6,4)

6=1x4+2, (6,4)=(4,2)=2

4=2x2, from which:

6=22-1x16

4=16-2x6=16-2x(22-1x6)=3x16-2x22

2=6-1x4=22-1x16-1x(3x16-2x22)=3x22-4x16=Ax22+Bx16

In this case n\(\displaystyle _{k}\) is 2

Now the question. Again fron Meserve:

"The equation (2-10) is both necessary and sufficient for n\(\displaystyle _{k}\) to be the greatest common divisor of m and n. It is necessary because of Theorem 2-11, and sufficient since if (2-10) holds, every common factor of m and n divides n\(\displaystyle _{k}\)."

What is this saying, ie, translate it into:

If "R" then "S".

If "S" then "R".

What are "R" and "S"