If N different positive integers are added and the sum is denoted as S, and if S=NX, is X an integer? (1) N is odd. (2) All N numbers are consecutive integers. Kudos for a correct solution .

statement 1: pick 1, 2, 3. Sum=6=3x and x is an integer. pick a set of odd prime numbers excluding 3 such as 7, 17, 19 or 13, 17, 5 sum is 35=3x and x is not an integer. statement 2: pick 1, 2, 3, 4 sum=10=4x and x is not an integer. pick 11, 12, 13 sum=avg(n)=36=3x and x is an integer. 1+2) the average of an odd number of consecutive integer is equal to its median. In an evenly spaced set with odd number of elements the sum will always be a multiple of the average, which is always going to be an integer because the number of elements in the set is odd. So x is our average/median and it is always going to be integer. Answer C.

VERITAS PREP OFFICIAL SOLUTION Statement (1): N is Odd. If the numbers are 1,2 and 3 then 6=3 * 2 here X is an integer. and if we take 2, 3, 5, then 10=3 * (10/3) here X is a fraction. Insufficient. Statement (2): All the N numbers are consecutive integers. If N is even, and if we take 1 and 2, then S=3 = 2 * 1.5 so X is a fraction. if N is odd, and if we take 1,2 and 3, then S = 6 = 3 * 2, so X is a integer. Insufficient. If we combine both the statements, then N is odd and all the numbers are consecutive integers, in that case X has to be positive integer. Hence Option C.

stmt 1: 1,2,3 6=3X; X=2 2,4,8 14=3X; X=Fraction NS stmt 2: 1,2,3 6=3X; X=2 7,8,9,10 34=4X; X= Fraction NS combine 1&&2 If N is odd and we have consecutive integers 2 possibilities: sequence starts with a EVEN integer eg: 2K + 2K+1 + 2K+2 = 6K+3 is divisible by 3 sequence starts with a ODD integer (2K-1) + (2K) + (2K+1) = 6K is divisible by 3 we can test this with N=4 as well. Answer : C

Hi Bunuel, could you please shed some light on the highlighted portion of your explanation for this question ? If we combine both the statements, then N is odd and all the numbers are consecutive integers, in that case X has to be positive integer. Hence Option C. How do we know this is the case ? ( I know this is the case and i solved it using examples 2k, 2k+1.. etc. not sure how did you get this ). Regards, Lucky

The question basically asks whether S (the sum of N integers) is divisible by N (the number of integers). Properties of consecutive integers: • If n is odd, the sum of n consecutive integers is always divisible by n. Given \{9,10,11\} , we have n=3=odd consecutive integers. The sum is 9+10+11=30, which is divisible by 3. • If n is even, the sum of n consecutive integers is never divisible by n. Given \{9,10,11,12\} , we have n=4=even consecutive integers. The sum is 9+10+11+12=42, which is NOT divisible by 4. (1) says that n is odd and (2) says that the numbers are consecutive, hence the sum of N integers, S, must be divisible by the number of untergers, N. Hope it's clear. Similar questions to practice: if-z1-z2-z3-zn-is-a-series-of-consecutive-positive-90974.html if-n-is-a-positive-integer-then-n-n-1-n-2-is-116251-20.html x-is-the-sum-of-y-consecutive-integers-w-is-the-sum-of-z-88044.html is-the-sum-of-the-integers-from-54-to-153-inclusive-divisib-160226.html is-k-2-odd-118591.html if-the-sum-of-5-consecutive-integers-is-x-which-of-the-148940.html the-function-f-m-is-defined-for-all-positive-integers-m-as-108309.html the-average-arithmetic-mean-of-the-even-integers-from-0-to-85907.html if-n-is-an-integer-greater-than-6-which-of-the-following-must-be-divi-100936.html is-k-2-odd-1-k-1-is-divisable-by-2-2-the-sum-of-k-95658.html are-i-j-k-consecutive-integers-101707.html if-a-b-c-are-consecutive-positive-integers-and-a-b-c-whic-108863.html if-n-is-an-integer-greater-than-50-then-the-expression-n-171237.html

The question basically asks whether S (the sum of N integers) is divisible by N (the number of integers). Properties of consecutive integers: • If n is odd, the sum of n consecutive integers is always divisible by n. Given \{9,10,11\} , we have n=3=odd consecutive integers. The sum is 9+10+11=30, which is divisible by 3. • If n is even, the sum of n consecutive integers is never divisible by n. Given \{9,10,11,12\} , we have n=4=even consecutive integers. The sum is 9+10+11+12=42, which is NOT divisible by 4. (1) says that n is odd and (2) says that the numbers are consecutive, hence the sum of N integers, S, must be divisible by the number of untergers, N. Hope it's clear. Similar questions to practice: if-z1-z2-z3-zn-is-a-series-of-consecutive-positive-90974.html if-n-is-a-positive-integer-then-n-n-1-n-2-is-116251-20.html x-is-the-sum-of-y-consecutive-integers-w-is-the-sum-of-z-88044.html is-the-sum-of-the-integers-from-54-to-153-inclusive-divisib-160226.html is-k-2-odd-118591.html if-the-sum-of-5-consecutive-integers-is-x-which-of-the-148940.html the-function-f-m-is-defined-for-all-positive-integers-m-as-108309.html the-average-arithmetic-mean-of-the-even-integers-from-0-to-85907.html if-n-is-an-integer-greater-than-6-which-of-the-following-must-be-divi-100936.html is-k-2-odd-1-k-1-is-divisable-by-2-2-the-sum-of-k-95658.html are-i-j-k-consecutive-integers-101707.html if-a-b-c-are-consecutive-positive-integers-and-a-b-c-whic-108863.html if-n-is-an-integer-greater-than-50-then-the-expression-n-171237.html Gotcha !! thanks .

I hope this will make the solution faster and simpler, but remember the magic formula of a sum of consecutive numbers: #numbers*(num_min+num_max)/2 1) clearly not suff: S can be the sum of any number 2) 1+2=3 -> 3/2 not an integer II 1+2+3= 6 -> 6/3 integer 1&2 Since number consecutive S=N*(n1+nN)/2 = NX Since N is odd n1+nN is even and so (n1+nN)/2= X = integer Suff.

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If N different positive integers are added and the sum is denoted as S : DS Retired Questions

If N different positive integers are added and the sum is denoted as S

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