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Question

Prove that 5 is irrational. [4 MARKS]

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Solution

Concept : 1 Mark
Application :1 Mark
Method : 2 Mark

Let us assume, to the contrary, that 5 is rational.

So, we can find co-prime integers a and b (0) such that 5=ab

5b=a Squaring on both sides , we get

5b2=a2 Therefore, 5 divides a2

Therefore, 5 divides a

So, we can write a=5c for some integer c.

Substituting for a, we get 5b2=25c2 b2=5c2

This means that 5 divides b2, and so 5 divides b.

Therefore, a and b have at least 5 as a common factor.

But this is contradiction to the assumption that 5 is rational.

So, we conclude that 5 is irrational.

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