2016 AMC 10A Problems/Problem 22
For some positive integer , the number has positive integer divisors, including and the number . How many positive integer divisors does the number have?
Since the prime factorization of is , we have that the number is equal to . This has factors when . This needs a multiple of 11 factors, which we can achieve by setting , so we have has factors. To achieve the desired factors, we need the number of factors to also be divisible by , so we can set , so has factors. Therefore, . In order to find the number of factors of , we raise this to the fourth power and multiply it by , and find the factors of that number. We have , and this has factors.