In a specific cryptographic protocol, a validation factor $V$ is calculated for prime numbers $p$ using the formula

Prove that for every prime number $p$ in the range $10<p<20$, the validation factor $V(p)$ is a multiple of 24.

A software engineer claims that for any non-negative integer $n$, the processing time $T(n)=n^2+3n+5$ microseconds is never a multiple of 2. Use proof by contradiction to show that this claim is correct.