Taken from the final question in a recently completed math contest:
The positive divisors of 21 are 1,3,7, and 21. Let S(n) be the sum of the positive divisors of the positive integer n. For example S(21)=1+3+7+21=32.
a) If p is an odd prime integer, find the value of p such that S(2p^2) = 2613
b) The consecutive integers 14 and 15 have the property that S(14) = S(15). Determine all pairs of consecutive integers m and n such that m=2p and n=9q for prime integers p,q > 3, and S(m) = S(n).
c)Determine the number of pairs of distinct prime integers p and q, each less than 30, with the property that S(p^3q) is not divisible by 24.
I've managed to answer both a, b however for the life of me I cannot begin to think of a way to prove c. The math contest is over however the fact that I cannot answer the last question has been bugging me ever since. Help please? Also please provide a explanation to your solution rather than just throwing out a number if possible. As I would really like to see the though processes behind how other's approached these questions.