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Question

If 3 is factor of p2, prove that 3 is a factor of p.

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Solution

We have,

If 3 is factor of p2.

Prove that:- 3 is factor of p

Proof:-First of all theorem holds only

If 3 is prime number.

So, Suppose that,

3 is factor of p2 but 3 is not divide p. since 3 is a prime

Then,

G.C.D. of (3,p)=1

Hence, their exists integers x and y such that,

3x+py=1

Since, 3 is a factor of p2

There exists a non zero integer n such that,

3x=p2

From equation (1) to,

(3x+py=1)×p

3xp+p2y=p

Replacing p2 by 3n

3px+3ny=p

3(px+ny)=p

This shows that, Which is a contradiction.

Hence, 3 is a factor of p.

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