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Where do you pull 2X3X3X5X11 out of? Obviously I see that equals 990, but where do you come up with that? Why not 5X9X22? Basically whats the process? can you explain some of the concepts for a guy who hasn't had math in years. If I get a problem like this I should..... and then give me a step by step. Pretend you are explaining it to a kid with a learning disability instead of your buddy from matrix algebra.
haha, yeah, that makes sense. Thanks. The other thing I'm still kind of wondering is that just because something is factor of both 990 and n! (11! for example) that doesn't necessarily mean that n! is a multiple of 990 right?
For example 12 is a factor of both 24 and 60, 60 is not a multiple of 24. Do I have make sure after figuring out that 11 is a factor of 990 that 11! is indeed a multiple of 990? I feel like I am missing another principle her.?
That 990 = 2x3x3x5x11 is the "prime factorization" of 990. Each of those numbers is prime, meaning that they cannot be divided by any number except themselves and 1. The "Fundamental Theorem of Arithmetic" says that each number is uniquely factorable into the product of primes. As the name suggests, this theorem is very useful (as hosam nicely showed in his solution).
Your factorization includes 9 and 22, neither of which is prime so it is not the "prime factorization". A really important use of the theorem for the GMAT is that if a number N is divisible by a number k that means that all the prime factors of k must also be in N. For example, suppose the question says find the smallest number k so that 270k is divisible by 33. This looks like a nightmare problem on the calculator, but using the FT it's easy.
270 = 3x3x3x2x5
33 = 3x11
so the answer to the question is that the product 270k needs to include an 11 and a 3. It does include a 3 already, so the answer is 11.
Thanks guys, this is really helping me get a grasp on this. One last question, how do you break things down so quick into their prime factors? Like 270 = 3x3x3x2x5. Is there a trick to this or are you just figuring out the prime factors of 27 and then adding 2X5?
I completely agree with hosam's answer. The only thing I would add is that there is no general way to get the prime factorization except coming up with a prime that divides the number and reducing the problem. So for the 270 you need to think of something that divides 270 and go from there. If the number is 15877, I don't have a solution either except to start trying numbers (and it might be prime). The GMAT would only have easily factored numbers like 270 for this kind of problem.