Bunuel wrote:
If x and y are both prime, is x*y = 323?
(1) x is the first prime number after 18
(2) y is the last prime number before 180
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:So the first thing that came to my mind is “Wow, that’s random”. The premise seems so arbitrary that it makes many approaches seem irrelevant. Even knowing that the two numbers are prime, we cannot quickly determine whether they must multiply to 323 without some more analysis and manipulation. Luckily, this is a Data Sufficiency question, so we have two additional statements that can help guide our analysis.
It’s important to note that in Data Sufficiency, we are trying to determine whether we can say with certainty that the two numbers multiply together to 323. This also means that if we can determine with certainty that the two numbers cannot multiply to 323, we have sufficient data. The uncertainty arises when we don’t know either way (i.e. maybe), so that provides a good framework for our analysis.
The first statement gives us a big hint, telling us that x is the first prime number after 18. This very quickly implies that x must be 19. We now have a hint as to why the number 323 was chosen (perhaps the author drove a Mazda in the ‘90s). If 323 is not a multiple of 19, then statement 1 will provide definitive evidence that x*y cannot possibly equal 323. Short of using a calculator, we can find multiples of 19 that are nearby and iterate manually until we find the correct answer. 19 x 20 would be easy to calculate as we can consider it as 19 x 2 x 10, or 38 x 10, or 380. From there, we can drop 19s until we get in the correct range.
380 – 19 is 361
361 – 19 is 342
342 – 19 is 323
You might be able to get there faster than by using this strategy, but after a few seconds of calculations, you can determine that 19 * 17 yields exactly 323. The question indicated that x and y would both be prime numbers, and 17 is indeed a prime number, so the possibility exists. However, it’s important to note that we know nothing (John Snow) about the value of y, other than it is a prime number. It could just as easily be 2, or 7, or 30203 (yes that’s a prime; I like palindromes). Since y could have any prime value, there’s insufficient evidence to determine that the product of x and y must be 323. Statement 1 is insufficient, and we can eliminate answer choices A and D.
Statement 2 indicates that y is the last prime number before 180, but it is important to remember that we must evaluate this statement alone. We now have no information about the value of x, other than it is a prime number. Statement 2 gives us a specific value of y, even if we’re not exactly sure what it is. We could do a little math and check to see if 179 (the number right before 180) is a prime, and in this case it is. The verification process is somewhat tedious, you have to check to see if it’s divisible by any prime number smaller than the square root of the number, so once you check 2, 3, 5, 7, 11 and 13, you’re confident than 179 is a prime number.
Knowing only that x is a prime number, we must now try and determine whether 179 and any prime could yield a product of 323, and the answer is very quickly no. The smallest prime number is 2, and 179 * 2 is already 358. You can also visually determine that 179 is more than half of 323, so there’s no need to even formally calculate the result. This statement on its own guarantees that x * y can never be 323, and thus is sufficient information to answer the question. The correct selection is answer choice B, as this statement alone is sufficient.
It is important to point out that these statements, taken together, give very clear numbers for both x and y. When this happens, you know that you can combine the statements and get only one value. That value may or may not be 323 (in this case it’s really, really not), but either way it provides sufficient information to definitively answer the question. However, it is almost always going to be the wrong answer, as it simply provides too much information. There’s no mystery or intrigue left, everything is laid out on the sheet in front of you. In business, as in life, if something seems too good to be true, it usually is.
Indeed, this question is essentially testing to see whether you’ll overpay for information on Data Sufficiency. However, at first blush, it just seems like an arbitrary collection of numbers with a question attached. When faced with similar head-scratchers, keep in mind that the statements (and/or answer choices) will provide hints. Trying to factor out 323 without any hints is a challenging endeavour, so look for hints and exploit them as much as possible. Hopefully, on test day, the only head scratching you’ll do is wondering which school you’ll go to with your outstanding score.
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