If x is the sum of n consecutive positive integers, and n2, is n

Question

If x is the sum of n consecutive positive integers, and n>2, is n divisible by 4?

(1) The smallest number among the consecutive integers is even.

(2) x is odd.

Quite a challenging one.

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Maria240895chile · Answer

(i) n=4 -> 2,3,4,5 = 14 n is divisible by 4 when the smallest is even.
 n=3 -> 2,3,4=9 n is not divisible by 4 when the smallest is even

NOT SUFFICIENT

(II) x is odd
in order to x to be odd, we need an odd N, and that always the set of numbers start with 1, and when that happen, n is always odd, hence, not a multiple of 4.

IMO B

gmatophobia · Answer

We can represent the series as

a \quad a+1 \quad a+2 \quad a+3 \quad - - - - - \quad a+(n-1)

Sum =  na + (1 + 2 + 3 + - - - + (n-1))

Sum =  na + \frac{n(n-1)}{2}

x =  n(\frac{ 2a + (n-1)}{2})

Let's start with Statement 2

Statement 2

(2) x is odd.

If x is odd, n is odd.

Is n divisible by 4 -- No !

The statement is sufficient and we can eliminate A, C, and E.

Statement 1

(1) The smallest number among the consecutive integers is even.

a = even

However, the statement doesn't tell us anything about the nature of n.

Hence the statement is not sufficient.

Option B

fdfd97 · Answer

Statement 1- First number is even, but we don't know how many terms. there are. So we can't determine if the total number of terms is div. by 4.

Statement 2- Sum of all the terms is odd. This implies that the number of terms is odd as well, so n can't be divisible by 4. So Option B.

TheBipedalHorse · Answer

If x is the sum of n consecutive positive integers, and n>2, is n divisible by 4

I'm having trouble with this but let me give it a try

let consecutive integers be a, a+1, a+2, .....

Sum(3 consec. integer) = 3a + 3
Sum (4 consec integer) = 4a + 7
Sum (5 consec integer) = 5a + 12

(1) The smallest number among the consecutive integers is even.
3 consecutive integers - a is even, a+1 is odd, a+2 is even, sum is odd
4 consecutive integers - a is even, a+1 is odd, a+2 is even, but a+3 is odd making the whole sum even

(2) x is odd.
that alone does not tell us the number of terms

but if smallest is even, and x is odd, then if we are talking about 6 consecutive integers, and 6 is not divisible by 4 but this occurs at every n being a multiple of 6
at every n = 4b+2 where b is a positive integer, sum (x) is odd
This is never divisible by n

(1) and (2) both are sufficient

chanhthang1802 · Answer

Question:  x = sum of n consecutive positive integers, n > 2, n/4 = integer?
 (1)  the smallest term is even
If the set is (2, 3, 4, 5) --> n = 4 --> the answer is Yes
If the set is (2, 3, 4) --> n = 3 --> the answer is No
Therefore, (1) is not sufficient
 (2)  x is odd
If n/4 = integer, the set will alternate between odd and even terms, with an even number of odd-even pairs. Sum of these pairs (each is odd) will be even.
For example, if n = 4 and the set is (1, 2, 3, 4), x = (1 + 2) + (3 + 4) = odd + odd = even
If n = 8 and the set is (1, 2, 3, 4, 5, 6, 7, 8), x = (1 + 2) + (3 + 4) + (5 + 6) + (7 + 8) = odd + odd + odd + odd = even.
Therefore, if n/4 = integer, x will never be odd. But statement (2) gives that x is odd, thus n can't be divisible by 4.
Statement  (2)  is sufficient. Answer is  (B) .

szcz · Answer

The statement 2 inference that the number of terms is odd is wrong. We can have the sum of all numbers being odd even with number of terms being even. 
For example: 4,5,6,7,8,9. Sum of all terms = 39. Number of terms = 6

Sum of all terms being odd implies that the number of terms may be odd or even(but not divisible by 4).

fdfd97 · Answer

Sorry, you are right. Then as per Statement 2, x has to be odd itself or consist of odd number of pairs. In either case, x isn't divisible by 4.

If x is the sum of n consecutive positive integers, and n>2, is n

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