Two positive integers a and b have HCF 12 and LCM 2520…

(a) For any two positive integers, HCF $\times$ LCM $= ab$. So $ab = 12 \times 2520 = 30240$.

(b) Put $a = 12x$ and $b = 12y$ with $\gcd(x, y) = 1$. Then $\operatorname{lcm}(a, b) = 12xy = 2520$, so $xy = 210 = 2 \cdot 3 \cdot 5 \cdot 7$. Each of the four primes goes into $x$ or into $y$, never both: $2^4 = 16$ ordered pairs.

(c) The sum $x + y$ is least when $x$ and $y$ are close. The coprime splits of $210$ are $1 \cdot 210$, $2 \cdot 105$, $3 \cdot 70$, $5 \cdot 42$, $6 \cdot 35$, $7 \cdot 30$, $10 \cdot 21$ and $14 \cdot 15$. The best is $14 + 15 = 29$, so the least $a + b = 12 \times 29 = 348$.

(d) The greatest comes from $1 \cdot 210$: $12 \times 211 = 2532$.

(e) One number divides the other only when $x = 1$ or $y = 1$, because $x$ and $y$ are coprime. That removes the two pairs $(12, 2520)$ and $(2520, 12)$, leaving $16 - 2 = 14$.