Prove that 3 divides 2n^2 + 1 if and only if 3 does not divide n

We have the operation:

(2n² + 1) / 3

2n (n/3) + 1/3

Since we are to use the condition that 3 does not divide n, we have:

n/3 = q + r/3

n = 3q + r

where q is the quotient and r is the remainder and not divisible by 3 or equal to 0

both q and r are whole numbers

Substituting,

2 (3q + r) (q + r/3) + 1/3

6q² + 4qr + 2r²/3 + 1/3

6q² + 4qr + (2r² + 1) / 3

The term:

(2r² + 1) / 3

will only be a whole number if r is not divisible by 3 or equal to 0, which means that

(2n² + 1) / 3

is a whole number if and only if

n/3 is not a whole number