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