Liouville’s Theorem: The Sum of the Squares of the Divisors of a Sum Equals the Sum of the Cubes of the Divisors of That Sum

I will verify the following, which appeared in Fermat's Library.
Liouville proved the following property of the divisors of the divisors of a number:
1) Start with a number (e.g., 10)
2) List its divisors (1, 2, 5, 10)
3) List the number of divisors for each divisor (1, 2, 2, 4)
4) (1 + 2 + 2 + 4)² = 81 = 1³ + 2³ + 2³ + 4³