Suppose that p epsilon Z If 3 divides p2 then 3 divides p Pr

Suppose that p epsilon Z. If 3 divides p^2, then 3 divides p. Prove that squareroot 3 is not rational.

Solution

1)

Contrapositive of above statement is

IF 3 does not divide p then 3 does not divide p^2

Let,3 not divide p

Let, p=3k+r, k is some integer and r=1 or 2

p^2=9k^2+r^2+6k=3(3k^2+2k)+r^2

r^2=1 or 4

HEnce, p^2 is not a multiple of 3

HEnce, 3 does not divide p^2

2.

Let, sqrt{3} be rational

So, sqrt{3}=p/q, p,q integers, gcd(p,q)=1

3q^2=p^2

Hence, 3|p^2

From Problem 1) 3|p

So, p=3k for some integer k

3q^2=3^2k^2

q^2=3k^2

Hence 3|q

But, gcd(p,q)=1

and we have,3|q,3|p

Hence a contradictin

Hence, sqrt{3} is irrational