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# Prove that is sqrt3 irrational number. - CBSE Class 10 - Mathematics

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#### Question

Prove that is sqrt3 irrational number.

#### Solution

Let us assume, to contrary, that  is rational. That is, we can find integers a and b (≠0) such that sqrt2=a/b

Suppose a and b not having a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.

So, bsqrt3= a

Squaring on both sides, and rearranging, we get 3b2 = a2.

Therefore, a2 is divisible by 3, and by Theorem, it follows that a is also divisible by 3.

So, we can write a = 3c for some integer c.

Substituting for a, we get 3b2 = 9c2, that is, b2 = 3c2.

This means that b2 is divisible by 3, and so b is also divisible by 3 (using Theorem with p = 3).

Therefore, a and b have at least 3 as a common factor.

But this contradicts the fact that a and b are coprime.

This contradicts the fact that a and b are coprime.

This contradiction has arisen because of our incorrect assumption that sqrt3 is rational.

So, we conclude that sqrt3 is irrational.

Is there an error in this question or solution?

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Solution Prove that is sqrt3 irrational number. Concept: Proofs of Irrationality.
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