Hello,

Let d=gcd(m,n) and d'=gcd(m+kn,n)

1st part

d divides m.

It also divides n.

So it divides every linear combination of the two, and in particular m+kn.

Therefore, d divides both m+kn and n.

We can conclude that d divides d' (because d' is thecommon factor)highest

2nd part

d' divides n.

d' also divides m+kn.

So it divides every linear combination of the two, and in particular m+kn+(-k)n=m.

Therefore, d' divides both m and n.

We can conclude that d' divides d.

----> d=d'