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February 29, 2020

Well ordering principle and Bezout Theorem

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Consider the following set of numbers:

$$ \displaystyle {M = \{ \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ... \} }$$

Does this set have a least number? Can you rigorously prove your answer?

Concepts in this lesson will help you to answer this question and more.

Concept - Well ordering principle, Bezout Theorem

The well-ordering principle states that every non-empty set of positive integers contains a least element.

Counter Example: The set of rational numbers does not have this property

Bezout Theorem: Let a and b be integers with greatest common divisor d. Then, there exist integers x and y such that ax + by = d. More generally, the integers of the form ax + by are exactly the multiples of d.

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