Two things I would say here:

One is that I think a big part of number property questions is just having good strategies on DS, since the questions appear more on DS and since they tend to lend themselves to picking numbers, etc. You often do not need to get too technical with the math and break everything down into primes on those questions. Often just good strategic number picking (with a healthy dose of DS know-how) is enough on these questions.

That said, I personally don't like ManhattanGMAT's prime box technique and prefer something that

Kaplan used to teach (not sure if they still do). I am not sure if I can do this justice in a post, but let me try. Basically the concept is that a multiple divided by a factor must equal an integer. If you think about that, it is obvious, but it is more about how you apply it. So on a question that asks is 10! a multiple of 2^6, 10! is the multiple and 2^6 is the factor. We don't know yet if 10! is a multiple of 2^6 or if 2^6 is a factor of 10!, but if you break both down into primes it is fairly easy to see - basically you are trying to see if 10! divided by 2^6 would result in an integer. After breaking 10! down you will see that there are 8 2's that will be in the numerator of that equation and of course 2^6 is just 6 2's. So there are enough 2's in the numerator to cancel out the ones in the denominator and if there is no denominator left then the result will definitely be an integer and thus 10! is a multiple of 2^6 and 2^6 is a factor of 10!. Basically you put the multiple on the top, factor on the bottom, and then break everything down into primes to see if you have enough in the numerator to cancel what you have in the denominator.

I am not sure which

Kaplan book you might find to give you a better explanation of that, but it is the technique that I use and the one that I teach to my students. Its not the only thing that you might use on divisibility and primes questions, but it would handle many of the tougher ones. But again to reiterate, often these questions, especially when they appear in DS, are really better approached by just good effective number picking.

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