Prove If 3 divides n2 then 3 divides nSolutionLet if possibl

Prove: If 3 divides n^2 then 3 divides n

Solution

Let if possible 3/n^2 but 3 does not divide n.

Then n² = 3k for some integer k

As 3k is a perfect square, if 3 is a factor then another 3 also should be factor

i.e. k =3m for some integer m.

i.e. n² = 9m² =3k

n = 3m

which implies that 3 divides m contrary to our assumption.

Hence our assumption was wrong

If 3 divides n^2 then 3 divides n