Problem:
Let . How many positive integer divisors of are less than but do not divide ?
Solution:
Let , where and are distinct primes. Then , so has
factors. For each factor less than , there is a corresponding factor greater than . By excluding the factor , we see that there must be
factors of that are less than . Because has factors (including itself), and because every factor of is also a factor of , there are
factors of that are less than but not factors of . When and , there are such factors.
The problems on this page are the property of the MAA's American Mathematics Competitions