Question

14. If p is a prime number, then prove that Vp is irrational.
Long Answer Type Question​

Answer 1

Step-by-step explanation:

Let us assume on the contrary that

p

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

p

=

b

a

∴p=

b

2

a

2

∴b

2

p=a

2

⇒p ∣ a

2

[∵p ∣ b

2

p]

∴p ∣ a

∴a=pc for some positive integer c.

Now, b

2

p=a

2

∴b

2

p=p

2

c

2

[∵a=pc]

∴b

2

=pc

2

⇒p ∣ b

2

[∵p ∣ pc

2

]

⇒p ∣ b

∴p ∣ a and p ∣ b

This contradicts that a and b are co-primes.

Hence,

p

is irrational.......

hope it helps u.....