Question

if p is a prime no. prove that
$\sqrt{p} \:is \: irrational$

Answer 1

Let us assume that the square root of the prime number

p

p

is rational. Hence we can write

p

–

√

=

a

b

p=ab

. (In their lowest form.) Then

p=

a

2

b

2

p=a2b2

, and so

p

b

2

=

a

2

pb2=a2

.

Hence

p

p

divides

a

2

a2

, so

p

p

divides

a

a

. Substitute

a

a

by

pk

pk

. Find out that

p

p

divides

b

b

. Hence this is a contradiction as they should be relatively prime, i.e., gcd

(a,b)=1

(a,b)=1