An Introduction to Mathematical Proofs by Nicholas A. Loehr

Author:Nicholas A. Loehr [Loehr, Nicholas A.]
Language: eng
Format: epub, pdf
ISBN: 9780367338237
Publisher: CRC Press
Published: 2019-09-18T00:00:00+00:00

4.6 GCDs and Uniqueness of Prime Factorizations

In the last section, we looked at specific examples where gcd(a, b) could be expressed in the form ax + by for certain integers x and y. Here we prove that such expressions always exist. We then use this result to obtain further facts about prime integers. In particular, we prove that the factorization of a positive integer into a product of primes is unique except for rearranging the order of the factors.

Proof of the Linear Combination Property of GCDs

We now prove the linear combination property of gcds for all integers a, b ≥ 0; an exercise asks you to generalize to the case where a or b could be negative.

4.51. Theorem on Writing GCDs as Linear Combinations. For all integers a, b ≥ 0, there are integers x and y such that gcd(a, b) = ax + by. Proof. To clarify the structure of the induction proof, we write the theorem statement in formal symbols as follows:

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