Nice work, everyone. Taking Narik's strategy a step further:
The product will be odd (odd * odd) and the sum will be even (odd + odd), so subtracting those two you'll end up with an odd number as the result.
How do we know to do this? Well, there's just too much calculation possible - when calculations seem to be involved and tedious, you should always think about using number properties as a way to shorten them. There are simply too many combinations of these numbers (5, 7; 5, 11; 7, 11; 13, 17; 11, 17; etc.) to try double-digit, odd-number multiplication and do the subtraction. If there's a way to simplify that process - and knowing that you have some odd and some even answer choices is another nice clue - you should give that a shot first.
Now we know that B and D are out, leaving A, C, and E. What to do? I see a few clues:
-Those choices are all pretty nicely spread out - 21, 119, and 231. It's very, very possible that at least one of the outer choices will not be possible.
-If we do want to plug in primes via trial-and-error, it makes sense to be methodical. Try to get 21 first, and if we can't then move on - if we start with smaller primes, they'll be easier to use and we may find patterns that will help.
If we do that, we'll find that the SMALLEST pairing of 5 and 7 yields:
5*7 - (5+7) = 35 - 12 = 23
23 is too large to be 21, and that's the smallest we can get, so it's eliminated.
Now...we know that our remaining choices are 119 and 231, and we've already recognized that there's a limited range to the set of potential values here. Trial-and-error dictates that we need to try bigger numbers anyway, so let's try the biggest to see if we can either eliminate 231 or recognize how close we are to that range (for example, we might notice that 13 * 17 gets us very close to 231, but reducing one of the variables would do the trick...). So, let's try 13 and 17:
13*17 - (13+17) = 221 - 30, and we know that even at the biggest this number can be, we won't reach 231. Therefore E is eliminated and the correct answer must be C - it's the only plausible choice left.
When problems seem to involve too many calculations, too extensive of calculations, or both, try to use number properties and/or test the range of the answer choices to find a more efficient way of answering the questions. Even if you can only eliminate a few or limit the range somehow, you'll still drastically reduce your workload and the time it takes to complete these.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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