I've just begun number theory and am having a lot of trouble with proofs. I think I am slowly grasping it, but would appreciate some clarification or any tips on the following please.(adsbygoogle = window.adsbygoogle || []).push({});

Show that if a and b are positive integers, then there is a smallest positive integer of the form a - bk, k [itex]\in[/itex] Z

The answer given is:

Let a and b be positive integers and let

S = {n:n is a positive integer and n = a - bk for some k [itex]\in[/itex] Z}

Now S is nonempty since a + b = a - b(-1) is in S. By the well ordering principle, S has a least element.

So, are they implying that k is the smallest element? or a - b(-1) as a whole is the smallest element?

I'm unsure as to how they are showing that it is the smallest element though. Given that either of the two options above would be S[itex]_{o}[/itex], what value would be the comparing s?

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# Homework Help: Number Theory WOP

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