I saw a proof on a textbook, but I don't understand why m and n should not both be even?(adsbygoogle = window.adsbygoogle || []).push({});

Prove p^2 =2 then p is irrational

(1) p^2 = 2

is not satisfied by any rational p. If there were such a p, we could write p = m/n

where m and n are integers that are not both even.

Then (1) implies

(2) m^2=2n^2

This shows that m^2 is even.

Hence m is even (if m were odd, m 2 would be odd),and so m^2 is divisible by 4.

It follows that the right side of (2) is divisible by 4,

so that n is even, which implies that n is even.

The assumption that (1) holds thus leads to the conclusion that both m

and n are even, contrary to our choice of m and n. Hence (1) is impossible for

rational p.

Thanks in advance

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# Question about simple analysis proof

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