Suppose that the positive integer n has the property that $\displaystyle n + \sigma(n)$ is divisible by 3.

if $\displaystyle n = pq$, where $\displaystyle p$ and $\displaystyle q$ are distinct odd primes with$\displaystyle p < q$, then p = 3 and $\displaystyle q \equiv 5 (mod 6$).

if $\displaystyle n = pq$ then

$\displaystyle n + \sigma(n) = pq + (pq+ p + q + 1) = 2pq + p + q + 1$

working modulo 3, p can take any of the values 0, 1, 2 but q can take only values 1 or 2 since $\displaystyle q > 3$ is prime.

Now, I can understand why p can only take values 0, 1, 2, but why can q only take values 1 or 2? q > 3 is prime? I don't understand what is meant by this. Obviously and $\displaystyle q > 3$ is not prime for all q, and if q is defined as a odd prime this is just a pointless statement, so what does it mean and why does it put restrictions on what values q can take modulo 3?