# Q1  Prove that   is irrational.

Let us assume  is rational.

It means  can be written in the form  where p and q are co-primes and

Squaring both sides we obtain

From the above equation, we can see that p2 is divisible by 5, Therefore p will also be divisible by 5 as 5 is a prime number.

Therefore p can be written as 5r

p = 5r

p2 = (5r)2

5q2 = 25r2

q2 = 5r2

From the above equation, we can see that q2 is divisible by 5, Therefore q will also be divisible by 5 as 5 is a prime number.

From (i) and  (ii) we can see that both p and q are divisible by 5. This implies that p and q are not co-primes. This contradiction arises because our initial assumption that   is rational was wrong. Hence proved that   is irrational.

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