GMAT Question of the Day - Data Sufficiency - Divisibility
If k is an integer is (k^2 - 7k + 12)(k+2)/4 an integer?
(1) k^2 - 1 is not divisible by 4
(2) k is divisible by 4
GMAT Question of the Day Solution
For this GMAT question of the day you might consider that any set of 4 consecutive integers will have a one integer that is divisible by 4. 1, 2, 3, 4 or 7, 8, 9, 10. You also have to consider integer sets that have the same properties as a consecutive set: 1, 2, 3, 8 is the same as 1, 2, 3, 4 and 4, 9, 10, 11 is the same as 8, 9, 10, 11. So the integers don't have to be consecutive provided that you have 4 integers that have different remainders when divided by 4. If you have that then it is guaranteed that one of the integers will be divisible by 4.
Start this GMAT question of the day by factoring. Why? Because factoring will give you another way of looking at the numbers and in this case the factoring is simple. (k-3)(k-4)(k+2). So now we know that we have three numbers that have different remainders when divided by 4. How do we know that? None of the numbers are a multiple of 4 apart. That means that they are not in the same divisibility row. 4, 8, 12, 16, 20 are all in the same row of remainder 0. 1, 5, 9 are all remainder 1. 2, 6, 10 are all remainder 2. 3, 7, 11 are all remainder 3. We don't know if k-3 is remainder 0, 1, 2 or 3 but we do know that k-3 has a different remainder when divided by 4 than k-4 does. Same thing with k + 2. Now we are looking for the statements to help narrow down whether the missing one in the patter, in this case k-1, is divisible by 4. If we can figure that out then we can answer the question. If this concept is confusing re-read this and try to pick some real numbers to test all of this out.
Statement (1) is a difference of squares and should be factored: (k-1)(k+1) if this isn't divisibly by 4 then neither k-1 nor k+1 are divisible by 4. We already figured out above that if we know the divisibility of k -1 then we can answer the questions. But just to break it down farther: This means that k - 3 isn't divisible by 4 because it has the same divisibility by 4 as k +1. Therefore k-4 or k+2 (because k-1, k-2, k-3, k-4 are consecutive integers and k + 2 has the same divisibility by 4 as k - 2) must be divisible by 4 so the answer is yes. Sufficient.
Statement (2) If k is divisible by 4 then k - 4 must be divisible by 4 because they are in the same remainder row in terms of divisibility by 4. Pick some numbers to verify this. Sufficient.