Is the integer n even?

[Max@Math Revolution]

[GMAT math practice question]

Is the integer n even?

1) There is a sum of n consecutive integers that is even.
2) [n/2] is an even number, where [n] is the greatest integer less than or equal to n.

[Max@Math Revolution]

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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.

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)
Since the sum of 2, 3, 4, 5 and 6 is 20, which is an even number, and n=5 is an odd number, the answer is sometimes 'no'.
Since the sum of 1, 2, 3, and 4 is 10, which is an even number, and n=4 is an even number, the answer is sometimes 'yes'.
Condition 1) is not sufficient since it does not yield a unique answer.

Condition 2)
If n = 4, then [n/2] = 2 is even, and n is even.
If n = 5, then [n/2] = 2 is also even, but n is not even.
Condition 2) is not sufficient since it does not yield a unique answer.

Conditions 1) & 2)
Since the sum of 2, 3, 4, 5 and 6 is 20, which is an even number, and n=5 is an odd number such that [n/2] = 2 is even, the answer is sometimes 'no'.
Since the sum of 1, 2, 3, and 4 is 10, which is an even number, and n=4 is an even number such that [n/2] = 2 is even, the answer is sometimes 'yes'.
Conditions 1) & 2) together are not sufficient, since they do not yield a unique answer.

Therefore, E is the answer.
Answer: E

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.

[deloitte247]

Here, we want to determine if the integer 'n' is even.

Statement 1: There is a sum of 'n' consecutive that is even.
When n=5, sum of n-consecutive integer that is even = 2+3+4+5+6=20; here, n = odd
When n=4, sum of n-consecutive integer that is even = 1+2+3+4=10; here, n = even,
The information provided is not enough to arrive at a definite answer. Therefore, Statement 1 is INSUFFICIENT

Statement 2: n/2 is an even number where n is the greatest integer less than or equal to n.
If n=4, then n/2 = 4/2 = 2 which is even and n = even
If n=5, then n/2 = 5/2 = 2.5 which is not even. n = odd
Thus, the information provided is not enough to arrive at a definite answer. Therefore, Statement 2 is INSUFFICIENT

Combining statement 1 and 2 together;
Since the sum of 2, 3, 4, 5 and 6 = 20; n=5 (odd) and 20 is an even number.
So, n/2 = 5/2 = 2.5 = odd

Since the sum of 1, 2, 3, and 4 = 20; n=4 (even) and 10 is an even number. Such that n/2 = 4/2 = 2 = even
From the expression result above, the combination of both statement 1 and 2 are INSUFFICIENT

Answer = option E