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fitzpratik
Posts: 224
01 Dec 2018, 20:55
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
59% (01:46) correct
41%
(02:01)
wrong
based on 77
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If X and Y are positive integer, what is the number of factors of A?
A. A= X*Y, X and Y has only one common prime factor
B. X and Y each have exactly three factors
A. A= X*Y, X and Y has only one common prime factor
B. X and Y each have exactly three factors
03 Dec 2018, 14:59
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fitzpratik
What's the source of this problem? I'm asking because it has a couple of "not GMAT like" characteristics.
Statement (2) states that X and Y each have exactly three factors. That tells us that X and Y are both perfect squares, since only perfect squares have an odd number of factors.
It also tells us that X and Y are each the square of a prime number. Squaring any other number (4^2 = 16, 6^2 = 36) gives you something with more than three factors.
So, X and Y have to be numbers like 2^2 = 4, 3^2 = 9, 5^2 = 25, 7^2 = 49, etc.
By itself, since this statement doesn't say anything about A, it's not sufficient!
Statement (1) says that A = X * Y, and X and Y only have one common prime factor. How many factors does A have? Let's try to take this to extremes. What if X = Y = 2? In that case, their only common prime factor is 2. A = 4, which has 3 factors (1, 2, and 4).
What if X and Y are much larger numbers, with lots of factors (as long as they only share one prime factor)? For instance, X = 2*3*5*7*11*13, and Y = 2*17*19*23*29. Their only common prime factor is 2, but A = X*Y will have a ton of factors (no point in counting, since it's Data Sufficiency).
So, this statement is also not sufficient.
Finally, put the two statements together. This is where the problem is a little vague. Since A and B both have to be perfect squares, and since they're both primes, each of them has a prime factorization of the form p^2. Plus, each of them has the three factors 1, p, and p^2. (p might be a different prime depending on whether you're looking at A or B.)
However, it's not clear whether 'common prime factor' refers to unique prime factors. (If this was a real GMAT problem, it might be phrased like this, to be as clear as possible: "There is exactly one prime number that divides evenly into both A and B.") If it doesn't refer to unique prime factors, then it's actually impossible to combine the two statements. If they share one prime factor (such as p), since they're both perfect squares, they must both have two copies of p, meaning that they have more than one common prime factor.
But, let's assume that we're talking about unique primes! In that case, A and B have to be equal. Their product is p^4, which has five factors: 1, p, p^2, p^3, and p^4. So, the statements together are sufficient.
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