How to prove that $a+b$ and $a-b$ are coprime when $a$ and $b$ are coprime and their parities are different?

Question

When a and b are coprime and their parities are different,

How to prove that $a+b$ and $a-b$ are coprime?

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score -2 · Answer 1 · answered Oct 30 '22 at 11:44

-2

let $d\in\{\text{primes}\}\cup\{1\}$ s.t. $d|a+b,a-b\implies d|(a+b)+(a-b)=2a$ and $d|(a+b)-(a-b)=2b$, additionally $d\not= 2 $ (because $a+b$ is odd by hypothesis) $\implies d|a,b \implies d=1$

answered Oct 30 '22 at 11:44

Bongo

1,320

Unnecessary to assume $d$ prime. – Gerry Myerson Oct 30 '22 at 12:30
Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Oct 30 '22 at 12:41

How to prove that $a+b$ and $a-b$ are coprime when $a$ and $b$ are coprime and their parities are different?

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