Some Interesting Problems

These are problems that are supposed to make you think. Follow the links to be guided to the solutions.

How many divisors of 33! are perfect squares?
The n! (read n factorial) function give the product of the integers 1 through n.
33! = 1 · 2 · 3 ··· 31 · 32 · 33 = 8,683,317,618,811,886,495,518,194,401,280,000,000. Finding perfect square divisors of this 37 digit number seems like an impossible amount of work, but finding the number of these factors is fairly easy using nothing harder than prime factoring, and you don't need to directly factor the huge number. Start by thinking about the prime factorizatiion of n! .