Question 85
For positive integers $a$ and $b$, $\operatorname{gcd}(a, b)$ denotes their greatest common divisor and $\operatorname{lcm}(a, b)$ their least common multiple. Determine the number of ordered pairs $(a, b)$ of positive integers satisfying the equation
$$
a b+63=20 \operatorname{lcm}(a, b)+12 \operatorname{gcd}(a, b).
$$
$$
a b+63=20 \operatorname{lcm}(a, b)+12 \operatorname{gcd}(a, b).
$$