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Prove that is an irrational number.

Asked by Topperlearning User 4th June 2014, 1:23 PM
Answered by Expert

Let us assume, on the contrary that is a rational number.

Therefore, we can find two integers a, b (b 0) such that

where a and b are co-prime integers.

Therefore, a2 is divisible by 5 then a is also divisible by 5. So a = 5k, for some integer k.

This means that b2 is divisible by 5 and hence, b is divisible by 5.

This implies that a and b have 5 as a common factor.

And this is a contradiction to the fact that a and b are co-prime.

So our assumption that is rational is wrong.

Hence,cannot be a rational number. Therefore, is irrational.

Answered by Expert 4th June 2014, 3:23 PM
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