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?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Proof that there exists a smallest positive linear combination?

Loading...

Similar Threads - Proof exists smallest | Date |
---|---|

I Proof of Existence of Tensor Product ... Further Question .. | Mar 17, 2016 |

I Proof of Existence of Tensor Product ... Cooperstein ... | Mar 11, 2016 |

Simple Proof for the existence of eigenvector | Mar 21, 2014 |

Proof that exists prime btw n<p<n! | Oct 11, 2012 |

Does the elegant proof to the Fermats last theorem exists? | Jul 3, 2012 |

**Physics Forums - The Fusion of Science and Community**