How many ordered pairs $(a,b)$ of positive integers satisfy the equation $a\cdot b+13=12\cdot \mathrm{lcm}(a,b)+7\cdot \gcd (a,b)$, where $\gcd (a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\mathrm{lcm}(a,b)$ denotes their least common multiple?