Question

if p is a prime number then prove that √p is irrational.​

Answer 1

Answer:

Let p be a rational number and

p

=

b

a

⇒p=

b

2

a

2

⇒a

2

=pb

2

∴p divides a

2

But when a prime number divides the product of two numbers, it must divide atleast one of them.

here a

2

=a×a

p divides a

Let a=pk ......(1)

(pk)

2

=pb

2

⇒p

2

k

2

=pb

2

⇒b

2

=pk

2

∴p divides b

2

But b

2

=b×b

∴p divides b

Thus, a and b have atleast one common multiple p

But it arises the contradiction to our assumption that a and b are co prime.

Thus, our assumption is wrong and

p

is irrational number.

Step-by-step explanation:

Hope this helped u friend