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Some Cubes

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

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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?

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Factorial Fun

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

Rarity

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What we were looking for in the problem Euclid's Algorithm and Musical Intervals was, if you like, a 'ratio of ratios' but we were not able to find that exactly. In that problem you are asked to find rational approximations for the 'ratio of ratios' using Euclid's algorithm. If the process terminates then you will have found an exact 'ratio of ratios' but generally the process does not terminate.

Here you are asked to prove that 'ratios of ratios' in this sense are (nearly) always irrational.