Question

If p is a prime no. Them prove that √p is irrational

Answer 1

Here is your answer

Answer 2

p is a prime no.

The prime no. are 2,3,5,7.......

To prove:- root p i.e., root 2 , root 3 etc.

Now let root 2 be rational no. so it is in the is of a/b where a and b is a co-prime no. it means there is no other 1 and itself.

a/b= 2

a= b root 2

Squaring on both side,

(a)^2=(b root2)^2

a^2= 2b^2................ (1)

If p is a prime no. and p divides a^2 then it divides a also where a is positive integer.

a=2c

Squaring on both side

a^2=4c^2

Substitute eq (1)

2b^2=4c^2

b^2=2c^2

If p is a prime no. and p divides a^2 thenp divides a , where a is positive integer.

We considered there is no other common factor than 1 and itself but there is . so root 2 is irrational

root 2 is prime no. so root p also irrational number

Proved