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.