Prove if a prime p divides ab, then p divides a or b (ILIEKMATHPHYSICS)

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This video is part of the "Intro to Higher Math" series I am making; it mainly references Daniel Velleman's book "How to Prove it" Third Edition.

Thanks and enjoy the video!

ILIEKMATHPHYSICS

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A slightly shorter proof, which doesn't rely on other results:

Given p|ab and GCD(a, b) = 1, show that p|a or p|b.

Suppose p|a, then we are done.

So now suppose p∤a. If p|b, then once again, we are done.

So, now suppose p∤a and p∤b.
So, a = kp + r₁ and b = mp + r₂,
where 1 ≤ r₁ < p and 1 ≤ r₂ < p.
So, ab = p(knp + kr₂ + nr₁) + r₁r₂

Now as p|ab, then p|r₁r₂, but this is impossible as both r₁ and r₂ < p and p ≠ r₁r₂, as otherwise p would be composite.

Hence we conclude that p∤a and p∤b is not possible, so p|a or p|b.

davidbrisbane

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