A theorem from number theory states that, if a and b are nonzero integers, then there exists a smallest positive linear combination of a and b.(adsbygoogle = window.adsbygoogle || []).push({});

This is my proof:

Let S be a set such that S = {w[itex]\in[/itex]Natural numbers : w=am+bn} , where a and b are positive integers, m and n are any integers, and w is by definition a linear combination of a and b.

Suppose S is nonempty. Then S is a subset of the natural numbers. Then by the Well Ordering Principle, S has a smallest (positive) element. Thus there exists a smallest positive linear combination of a and b.

Is this correct? Or am I missing something? My professor said that fastest way to prove this is by contradiction, but it seems to me that just directly proving by the well ordering principle is faster?

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# Proof that there exists a smallest positive linear combination?

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