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**Th3sandm4n** The problem:

Let a,b,n be integers. n >0, such that n | (a-b) [n divides (aib)].

Show that (n,a) = (n,b) [(x,y) <==> the GCD of x and y].

Definitions

a | b means that b = ak + r for some integer r,k k=0;

Lemmas

Let a,b be integers. If a = bq+r, then (a,b) = (b,r).

Proof:

Since n | (a-b)

a-b = nk for some integer k.

a = nk + b

Using Lemma, (a,n) = (n,b).

I'm just wondering if there are any flaws in this proof, given I can use the lemma? I just want to make sure, seems a bit easy but my teacher is very strict and was just wondering if any of you could spot some possible criticisms.

Thanks