Sunday, 17 April 2016

elementary number theory - Prove sqrt2k is irrational where k is an odd integer.

Question:



Prove 2k is irrational where k is an odd integer.



My attempt:




Proof by contradiction:



Now, assume 2k is rational. Then, 2k=ab where a,bZ, b not equal 0 and a,b have no common factors.



2k=ab2k=a2b2(b2)(2k)=a22|a22|a, since 2 is primecZ such that a=2c



Then, (2k)(b2)=a2(2k)(b2)=4c2kb2=2c22|kb22|b22|b, since 2 is prime.



So, 2|a and 2|b , a contradiction.

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