Find all integers $m$ and $n$ with greatest common divisor $18$ and least common multiple $720$.

Recall that the product of the least common multiple and the greatest common divisor of two integers is given by

$$\textrm{lcm}(m,n) \cdot \gcd(m,n) = mn$$

As such, the product of the particular value of $m$ and $n$ we seek must be $18 \cdot 720 = 2^5 \cdot 3^4 \cdot 5$.

If we consider the prime factorization of $m$, note that it must contain at least $2 \cdot 3^2$, as this is the gcd of $m$ and $n$. If it contains more than a single $2$ in its factorization, it must contain $4$ of them, as otherwise, the gcd of $m$ and $n$ would be increased. Similarly, if it doesn't contain any more than a single $2$, $n$ has to contain the rest.

Likewise, either $m$ or $n$ (but not both) must contain the $5$ in their prime factorization. This gives essentially two possibilities:

  • One number could be $(2 \cdot 3^2) \cdot 2^3 \cdot 5$, leaving the other to be $(2 \cdot 3^2)$, or
  • One number could be $(2 \cdot 3^2) \cdot 2^3$, leaving the other to be $(2 \cdot 3^2) \cdot 5$

Multiplying things out, this produces the only possible pairs of values $\{720,18\}$ and $\{144,90\}$.