Bunuel wrote:
If a, b, c, and d are each integers greater than 1, is the product abcd divisible by 6?
(1) acd is even.
(2) abd is odd.
MANHATTAN GMAT OFFICIAL SOLUTION:Another way of saying that an integer is divisible by 6 is to say that it is a multiple of 6. Multiples of 6 must have all of the prime factors of 6 (6 = 2 × 3) and could have additional prime factors. Thus, our rephrased question is “Does the product abcd have at least one 2 and one 3 as prime factors?”
(1) INSUFFICIENT: This tells us that at least one of the integers a, c, and d must be even. Thus we have at least one 2 as a prime factor. However, we do not know anything about the remaining factors, and cannot determine whether there is one 3 among the prime factors of a, b, c, and d.
(2) INSUFFICIENT: This tells us that a, b, and d are all odd, which means there is no factor of 2 among their prime factors. Without information about c, we are uncertain about whether abcd has a factor of 2. Additionally, we have no information about the number of 3s among the prime factors of a, b, and d. It is possible that abd is 105, for example, and we would have the 3 required for divisibility by 6. On the other hand, abd could be 125 and we would have no 3s as factors.
(1) AND (2) INSUFFICIENT: If acd is even and if abd is odd, it must be true that c is even and that abcd has at least one factor of 2. Neither of the statements gives us a conclusive answer about the number of 3s among the prime factors of a, b, c, and d, however, and combining the statements does nothing to resolve that uncertainty.
The correct answer is E. _________________