Question

Prove that √7 is an irrational number.​

Answer 1

Let us assume that -/7 is rational.

Then, there exist co-prime positive integers

a and b such that-/7 =b/a

a ⟹a=b/7

Squaring on both sides, we geta /2 =7b^2

Therefore, 2 is also divisible by 7 and

so, b is also divisible by 7.

Therefore, a and b have at least one common factor, i.e., 7.

But, this contradicts the fact that a and b are co-prime.

Thus, our supposition is wrong.

Hence, -/7 is irrational.

Answer 2

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Let us assume that ^/7

is rational. Then, there exist co-prime positive integers a and b such that

7=ba

⟹a=b/7

Squaring on both sides, we get

a2=7b2

a2=7b2

Therefore, a2 is divisible by 7 and hence, a is also divisible by7

so, we can write a=7p, for some integer p

Substituting for a, we get 49p2

=7b2

⟹b2 =7p2.