Some Cubes

The sum of the cubes of two numbers is 7163. What are these numbers?

Em'power'ed

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?

Factorial Fun

How many divisors does factorial n (n!) have?

Rarity

Stage: 5 Challenge Level:

The proof depends on the prime factorizations of $a$, $b$, $c$ and $d$.