pzazz12 wrote:A number N when expressed as product of prime factors gives N= 2^5 * 3^2* 5^4 * 7 * 11^3. Find the number of ways the number N can be expressed as a product of three factors such that the factors are pair wise co-prime.
A.1027
B.3125
C.243
D.729
E.None of these
This is not a GMAT question; you don't need to know what 'coprime' means, and you won't see anything similar on the real test. It's also badly worded; the question does not make clear whether the order of the factors we are multiplying is important. We can, for example, write 15 as 3*5 or as 5*3; would these be counted as two different ways to express 15 as a product in this question, or as just one way?
So it's not a good question, and this solution is for interest only - if you're preparing for the GMAT, you can ignore it: two numbers are 'coprime', or 'relatively prime', if their GCD is 1. So 15 and 16, for example, are relatively prime. Here, if we assume the order of our factors matters, and if we want a*b*c = N, and a, b and c have no prime factors in common, then only one of them can be divisible by 2, so only one of them can be divisible by the 2^5 we need, and we have three choices for which of our letters will be divisible by 2^5. Similarly, only one of them can be divisible by 3, so we have three choices for which of our letters will be divisible by 3^2, and so on. So for each prime, we have three choices, and the answer will be 3^5 = 243.
