Question 23
How many ordered pairs $(a, b)$ of positive integers satisfy the equation
$$
\begin{aligned} a \cdot b + 63 &= 20 \cdot \operatorname{lcm}(a, b) \\ &\quad {}+ 12 \cdot \gcd(a, b), \end{aligned}
$$
where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b,$ and $\operatorname{lcm}(a, b)$ denotes their least common multiple?
$$
\begin{aligned} a \cdot b + 63 &= 20 \cdot \operatorname{lcm}(a, b) \\ &\quad {}+ 12 \cdot \gcd(a, b), \end{aligned}
$$
where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b,$ and $\operatorname{lcm}(a, b)$ denotes their least common multiple?