Hmm, something didn't go right with your method, because the answer should be 23 (assuming we are limiting factor pairs to the realm of positive number.) But PLEASE tell me this is not an OG problem, because I would FLIP OUT if something as obscure as this ever showed up on the GMAT. I *highly* doubt it would.
Here's the only way I know (though there may be others I'm not aware of) to go about the problem that doesn't just involve making a giant list (which would take way longer than any GMAT problem would ever be allowed to). It's actually not crazy difficult, I just can't imagine the GMAT would expect people to know or deduce this.
But, it is this:
You can find the number of factors (not factor pairs, just factors) a number has by thinking of the following (I'll use your 7056 for an example). Write out the prime factorization in exponential form. Here we have 7056 = 3^2 * 2^4 * 7^2. Then take a look at those exponents. What we learn from them is that any factor of 7056 will have
either zero, one, or two 3s;
either zero, one, two, three, or four 2s; and
either zero, one, or two 7s.
So that's
3 possibilities for how many 3s there are;
5 possibilities for how 2s there are; and
3 possibilities for how many 7s there are.
(In other words, the get how many different possibilities there are for how many of a particular number there could be, just add one to that number's exponent to account for the fact that there could be 0 of it.)
Now this becomes just like a counting problem where you have e.g. 3 possible shirts, 5 possible pairs of pants, and 3 possible ties, and you want to calculate how many different outfits you could form (only here, "outfits" = "factors"). So there are (3)(5)(3) factors, since every time you change your choice for how many of a certain prime factor there are, you create a new overall factor. So that's 45 factors total.
Now, normally, if you were looking for how many factor pairs there were, you'd just divide that number by 2, since each of the factors in the first half of the list will pair up with a factor in the later half, so we don't want to count those later-half factors as beginnings of new pairs. But clearly if we tried that here, we'd wind up saying there were 22.5 factor pairs, which makes no sense, since you can't really have half a pair. So, this case has one other special element, which is that 7056 is a perfect square, as we can tell from its prime factorization (i.e. we've got (3*2*2*7)*(3*2*2*7)) OR from the fact that it has an odd number of factors, a characteristic unique to perfect squares. It's winding up with an odd number of factors because if we listed those 45 factors in ascending order, that very middle number in the list -- the 23rd term out of the 45 -- would simply be "pairing" with itself (this happens with the "middle" factor for any square). So we really wind up forming 22 pairs by connecting the last factor with the first factor, the second-to-last with the second, the third-to-last with the third and so on, and then when the middle term is left alone, we count an additional pair for it, since it can pair with itself. So here you wind up with 23 factor pairs.
I kind of think this is an awesome trick, but again, if it showed up on the actual GMAT, I'd be shocked!
Ashley Newman-Owens
GMAT Instructor
Veritas Prep
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