, 2021

(a) By considering their difference, or otherwise, find all possibilities for the common factors of $n$ and $n + 3$. (1 mark)

For $n \geq 2$, let $P(n)$ denote the largest prime factor of $n$.

(b) If $a$ and $b$ are positive integers greater than $1$, explain why $P(ab)$ must be equal to at least one of $P(a)$ or $P(b)$. (1 mark)

(c) Find all positive integers $n$ such that $P(n^2 + 2n + 1) = P(n^2 + 9n + 14)$. (8 marks)