proof with integers q2?

Question

Suppose that n is an integer such that n^2 is divisible by 3. Prove that n is divisible by 3.
What do we know and what is required to be proven? how do i do this?


can u set it out with least writing as possible like explain things but without words with symbols and stuff more the mathematical working

Answer 1

Easier to prove that if n is NOT divisible by 3 then n^2 is not divisible by 3, they are equivlalent statements.

If n is not divisible by 3 then n=3k+x where x=1 or 2.
n^2 = (3k+x)^2 = 9k^2 + 6kx + x^2 = 3(3k^2+2kx) + (1 or 4)
= 3j + 1 for some integer j

Therefore if n is not divisible by 3, then n^2 is not divisible by 3.

Answer 2

Allison, I think this needs a negative proof. Basically, you show that if this isn't so, something that should be so won't be so.
Lets assume that n cannot be divided by 3 and produce an integer. Then n/3= A+r, where A is an integer and r is a remainder in thirds. Now if we multiply n/3 x n, we have nA+ nr. Since nr must be a integer (a given), nr must be divisible by 3.
Since r is either 1/3 or 2/3, for nr to be divisible by 3, n must be a multiple of 3. This contradicts our initial assumption, which then is not so. The alternate, that n is divisible by 3 is thus proven.