If x is an integer, is x³- 3x² + 2x divisible by 4?
(1) x = 3y, where y is an integer
(2) x = 7z, where z is an integer
Ans: A
Source: Derived from a DS problem in MGMAT
If possible, please tell me: What is the likely GMAT-difficulty-level of this Question?
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Hi npalani07,
This DS question has a quirky element to it which you really have to deal with first before you deal with the two Facts.
X^3 - 3X^2 + 2X can be factored into....
X(X^2 - 3X + 2) which can be further factored into....
X(X-1)(X-2)
So, the question ultimately asks: Is X(X-1)(X-2) evenly divisible by 4? This is a YES/NO question. We're told that X is an integer.
The "secret" to this question is that if X OR (X-1) OR (X-2) is divisible by 4, then the answer is YES. If NONE of them are divisible by 4, then the answer is NO.
Fact 1: X = 3Y; Y is an integer
This tells us that X is a multiple of 3 (0, 3, 6, 9, etc.)
If X = 0, then (0)(-1)(-2) = 0 and the answer is YES
If X = 3, then (3)(2)(1) = 6 and the answer is NO
Fact 1 is INSUFFICIENT
Fact 2: X = 7Z; Z is an integer
This tells us that X is a multiple of 7 (0, 7, 14, 21, etc.)
If X = 0, then (0)(-1)(-2) = 0 and the answer is YES
If X = 7, then (7)(6)(5) = 210 and the answer is NO
Fact 2 is INSUFFICIENT
Together, we know that X has to be a multiple of 3 AND 7 (0, 21, 42, etc.)
If X = 0, the the answer is YES
If X = 21, then (21)(20)(19) has 20 as a factor, so it IS divisible by 4 (YES answer)
If X = 63, then (63)(62)(61) has NO factor divisible by 4 (NO answer)
Together, INSUFFICIENT
Final Answer: E
These individual concepts can (and likely will) show up on the GMAT, but they're not necessarily going to show up "packaged" this way.
GMAT assassins aren't born, they're made,
Rich
This DS question has a quirky element to it which you really have to deal with first before you deal with the two Facts.
X^3 - 3X^2 + 2X can be factored into....
X(X^2 - 3X + 2) which can be further factored into....
X(X-1)(X-2)
So, the question ultimately asks: Is X(X-1)(X-2) evenly divisible by 4? This is a YES/NO question. We're told that X is an integer.
The "secret" to this question is that if X OR (X-1) OR (X-2) is divisible by 4, then the answer is YES. If NONE of them are divisible by 4, then the answer is NO.
Fact 1: X = 3Y; Y is an integer
This tells us that X is a multiple of 3 (0, 3, 6, 9, etc.)
If X = 0, then (0)(-1)(-2) = 0 and the answer is YES
If X = 3, then (3)(2)(1) = 6 and the answer is NO
Fact 1 is INSUFFICIENT
Fact 2: X = 7Z; Z is an integer
This tells us that X is a multiple of 7 (0, 7, 14, 21, etc.)
If X = 0, then (0)(-1)(-2) = 0 and the answer is YES
If X = 7, then (7)(6)(5) = 210 and the answer is NO
Fact 2 is INSUFFICIENT
Together, we know that X has to be a multiple of 3 AND 7 (0, 21, 42, etc.)
If X = 0, the the answer is YES
If X = 21, then (21)(20)(19) has 20 as a factor, so it IS divisible by 4 (YES answer)
If X = 63, then (63)(62)(61) has NO factor divisible by 4 (NO answer)
Together, INSUFFICIENT
Final Answer: E
These individual concepts can (and likely will) show up on the GMAT, but they're not necessarily going to show up "packaged" this way.
GMAT assassins aren't born, they're made,
Rich
