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Math Expert V
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Is the positive integer N divisible by 4?  [#permalink]

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Question Stats: 79% (00:56) correct 21% (01:31) wrong based on 34 sessions

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Is the positive integer N divisible by 4?

(1) N is a product of two even integers.
(2) N is a product of three consecutive integers.

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Re: Is the positive integer N divisible by 4?  [#permalink]

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Bunuel wrote:
Is the positive integer N divisible by 4?

(1) N is a product of two even integers.
(2) N is a product of three consecutive integers.

Target question: Is the positive integer N divisible by 4?

Statement 1: N is a product of two even integers.
Let x and y be the two even integers such that N = xy
Since x is EVEN, we can write x = 2k (for some integer k)
Likewise, since y is EVEN, we can write y = 2j (for some integer j)

This means, we can write: N = xy = (2k)(2j) = 4xy
Now that we have rewritten N as a multiple of 4, we can see that N must be divisible by 4
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: N is a product of three consecutive integers
This information is not sufficient. Consider these two conflicting cases:
Case a: N = (2)(3)(4) = 24. In this case, the answer to the target question is YES, N is divisible by 4
Case b: N = (1)(2)(3) = 6. In this case, the answer to the target question is NO, N is not divisible by 4
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Cheers,
Brent
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Is the positive integer N divisible by 4?  [#permalink]

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Bunuel wrote:
Is the positive integer N divisible by 4?

(1) N is a product of two even integers.
(2) N is a product of three consecutive integers.

Statement 1:

N is a product of two even integers, from each integer we can pull out a factor of 2. Since there are two even integers, we can pull out two factors of 2 which creates a 4, thus we are able to pull out a factor of 4 from N. Sufficient.

Statement 2:

We are checking if N is divisible by 4 and $$N = i * (i + 1) * (i + 2)$$ where $$i$$ is any positive integer. Then we want to separate into 4 cases of $$i$$ since we are in the divide by 4 world. We can try the cases {1 2 3}, {2 3 4}, {3 4 5}, and {4 5 6}. Each of them represent one of the cases: i = Multiple of 4 plus 0, 1, 2 or 3.

The first case is already not divisible by 4 while the others are, so this is insufficient. We can use this as a good example and see how {1 2 3} represents the entire case with $$i =$$ multiple of 4 plus 1. {5 6 7}, {9 10 11} are not divisible by 4 for example.

Ans: A
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Re: Is the positive integer N divisible by 4?  [#permalink]

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Bunuel wrote:
Is the positive integer N divisible by 4?

(1) N is a product of two even integers.
(2) N is a product of three consecutive integers.

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 1 variable (N) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)

We have N = (2a)(2b) = 4ab for some integers a and b.
Thus N is divisible by 4.

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

If N = 2*3*4 = 24, then N is divisible by 4 and the answer is "yes".
If N = 1*2*3 = 6, then N is not divisible by 4 and the answer is "no".

Since condition 2) does not yield a unique solution, it is not sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________ Re: Is the positive integer N divisible by 4?   [#permalink] 29 Feb 2020, 10:23
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