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subhajeet wrote: "HI Mike, I agree with your above post that every number has its own property. But my question is that if X = 33(A+B+C) => X is divisible by both 3 and 11. So why are we not taking into account that also???"
That's a great question. If X - 33*(something), then X we absolutely know that X must be divisible by 3 and 11. I used the divisibility by 3 in my solution, but not the divisibility by 11. Why?
There are a couple ways to answer that question. One is: the GMAT
definitely expects you to be able to look at a large number and tell whether it's divisible by three. That is one that I think I've seen at least on every full-length GMAT I've taken or studied. By contrast, divisibility by 11 -- yes, there is a relatively simple trick for that as well, but I have
never seen a real GMAT question where the test expected we would know that. So, that's part of the answer: divisibility by three is highly GMAT-relevant, and divisibility by 11 isn't at all.
The deeper answer has to do with the peculiarity of mathematical problem-solving. The route I happened to see to the answer to the original question was
algebraic formulation --> divisibility by 3 ----> divisibility by 9 ---> multiples of 9 times 33 ----> few enough to check one-by-one
Is that the best way to answer this question? I don't know. It's the best of which I could conceive. As it happens, in this route, divisibility by three was vital, and divisibility by 11, while perfectly true, was completely irrelevant. I point out, though, it's not surprising that the GMAT-practice source would include a question in which divisibility by 3 plays a vital role in the solution, since that's an important concept in the GMAT Math.
Perhaps another way to say it: in doing math at this level, we need to distinguish between (a) what's mathematically allowed, mathematically legal, vs. (b) what's
strategic: what will move me closer to the answer? Think about a very simple algebra equation: Given 2x + 5 = 13, solve for x. There are an infinity of mathematically legal steps we could take, but most of them would be nonstrategic. For example, I could begin by adding 317 to both sides of the equation: that is absolutely legal mathematically, but in terms of strategy it would be a completely daft move. Just because you
can do something, just because something is mathematically legal, does not mean that it's strategic, does not mean that it's a move that will pay dividends in getting you that much closer to the answer. In this problem, it would have been perfectly mathematically legal to explore divisibility by 11 instead of divisibility by 3, but so far as I can tell, the former path doesn't really go anywhere, whereas, the latter path leads elegantly to the solution.
Of all the things that are mathematically legal, how do you determine what's most strategic? There's no short answer to that one. Nothing replaces experience with problem-solving. There is a classic, relatively dense text by the mathematician George Polya,
How to Solve It, if you want to have a more theoretical introduction. (This pdf,
http://furius.ca/cqfpub/doc/proofs/how-to.pdf, gives the gist of Polya's approach). But, fundamentally, math is not a spectator sport: you learn it, you become better at it, only by doing it.
I hope, at least to some extent, that answers your question. Please let me know if you have questions on anything I have said.
Mike
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Mike McGarry
Magoosh Test Prep