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Question

Prove that 3 is an irrational number.

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Solution

Let us assume on the contrary that 3 is a rational number.

Then, there exist positive integers a and b such that
3=ab where, a and b, are co-prime i.e. their HCF is 1
Now,
3=ab
3=a2b2
3b2=a2
3 divides a2[3 divides 3b2]
3 divides a...(i)
a=3c for some integer c
a2=9c2
3b2=9c2[a2=3b2]
b2=3c2
3 divides b2[3 divides 3c2]
3 divides b...(ii)

From (i) and (ii), we observe that a and b have at least 3 as a common factor. But, this contradicts the fact that a and b are co-prime. This means that our assumption is not correct.

Hence, 3 is an irrational number.

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