Question 56
Call a positive integer $n$ practical if every positive integer less than or equal to $n$ can be written as the sum of distinct divisors of $n$.
For example, the divisors of $6$ are $\{1, 2\}$, $\{3\}$, and $\{6\}$. Since
$$
1=\{1\}, \quad 2=\{2\}, \quad 3=\{3\}, \quad 4=\{1\}+\{3\}, \quad 5=\{2\}+\{3\}, \quad 6=\{6\},
$$
we see that $6$ is practical.
Prove that the product of two practical numbers is also practical.
For example, the divisors of $6$ are $\{1, 2\}$, $\{3\}$, and $\{6\}$. Since
$$
1=\{1\}, \quad 2=\{2\}, \quad 3=\{3\}, \quad 4=\{1\}+\{3\}, \quad 5=\{2\}+\{3\}, \quad 6=\{6\},
$$
we see that $6$ is practical.
Prove that the product of two practical numbers is also practical.