For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75 = 3 x5x5. How many two digit positive integers have length 6?

A)None
B)One
C)Two
D)Three
E)Four

Can you please a) show the strategy to do this in less than 2 min and describe the range of difficulty of this problem?

I chose B (the incorrect answer) since 2^6 = 64 But the correct answer is C.

You need to use a little bit of logic to bound things here so that you can get through this in 2 minutes. I'd guess it's a 650-l700 evel question.

They give us the boundaries; we just have to understand what they mean. First, I'm only using 2-digit positive integers, so 10 to 99 inclusive. Second, I'm only using prime numbers. Third, I need to have a length of 6, so I need to multiply 6 prime numbers together.

The smallest prime number is two, so the first thing I try is six 2's (2*2*2*2*2*2) which equals 64. I know I can't get anything smaller than that, because two is the smallest prime. My next thought, then, is how to create the next-smallest possibility. To do that, I want to keep as many 2's as possible, but I have to change at least one of them to get a different product. The next smallest prime after 2 is 3, so I substitute a 3 for one of the 2's (2*2*2*2*2*3) which equals 96. That's still within the stated boundaries of the problem.

The next smallest possibility can be calculated by replacing one more 2 with a 3. If I think about it, I can tell that doing this will put me over 99 (though if I'm not sure, I can do the math to check) - so I'm done. There are two possibilities.

The key is to think about this logically and within the boundaries so that you know you've found all of the possibilities.

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

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they are.

incidentally, it's fundamentally important that you grasp this mentality: they are "a crafty bunch", and the "clever" aspect of gmat problems should not be underestimated (especially on data sufficiency problems).

if you plod through gmat problems trying to solve them as if they were "normal homework for school", you are not going to have a good time, to say the least.