The number of positive integers which divide 2004 to leave a remainder of 24 is?

Well, well, well, looks like we have a math problem on our hands. The number of positive integers that divide 2004 to leave a remainder of 24 is the same as the number of positive integers that divide 1980. Why? Because 2004 - 24 = 1980. So, the answer is the number of divisors of 1980, which can be calculated by prime factorizing 1980 and using the formula for the number of divisors. Voila!