Which of the following CANNOT be true if the sum of k consecutive inte

Question

Which of the following CANNOT be true if the sum of k consecutive integers is 0, where k ≥ 1 ?

I. The product of the k integers is positive
II. The smallest of the k integers is zero
III. The largest of the k integers is negative

A. I only
B. II only
C. III only
D. I and III
E. I, II and III

Lampard42 · Answer

I) where K=1 then number must be 0 which is neither positive nor negative; where K>1 then K must be Odd eg.3 -1,0,1 and so on where ) will be one of the terms causing the product to be zero - Cannot be true
II)Smallest of K is zero; Where K=1 and zero is the only term - can be true
III) Largest of K is negative then sum will never be zero - Cannot be true. 
IMO D

IanStewart · Answer

If the sum of the integers is zero, then their average must be zero. But it's a set of consecutive integers, so it's an equally spaced set, and the average equals the median. So the median must also be zero, and we have a zero in the middle of our set with an equal number of negative and positive integers on either side of it.

Since zero is in the set, the product of the integers in the set is zero, and I cannot be true.

Since the question allows us to have just one number in our set, our set could just be "0", and II could be true (but only in that one case)

Since zero is in the set, it's impossible for the largest value to be negative, because zero is bigger than any negative number. So III cannot be true.

CrackverbalGMAT · Answer

This is a moderately difficult question on numbers and inequalities. In this question, we need to find out which statement is ALWAYS FALSE.

Since the integers are consecutive which add up to ZERO, one of them HAS to be ZERO. Also, the other numbers will have the same magnitude but opposite signs. For example, the numbers can be -1, 0 and 1; or they can be -2,-1,0,1 and 2.

Also remember that k can be equal to 1 i.e. we can have 1 consecutive integer. So, we can take 0 as the only number in the list.

If we take the case of the numbers being -1,0 and 1:

Statement I is false since the product is 0.

Since Statement I is false, we can eliminate the options not containing statement I. Options B and C can be eliminated. We are left with A, D and E.

If we take the case of the number being 0,

Statement II is true but statement III is false.

So, option E can be eliminated.

It’s obvious that the largest of the k integers cannot be negative. As we saw in the examples, it has to be ZERO or positive, both of which are non-negative.

The correct answer therefore, has to be option D.
Hope this helps!

IanStewart · Answer

The product of -1, 0 and 1 is not equal to -1, and even if it were, I would be false in that case, not true.

CrackverbalGMAT · Answer

Hey Ian,

Thanks for pointing this out. It was a careless human error on my part. 
I have corrected the mistake and edited my response as well.

Regards

CareerGeek · Answer

Examples of set when sum of k consecutive integers is 0 are {0}, {-1, 0, 1}, {-2, -1, 0, 1, 2} etc

I. The product of the k integers is positive
 {0} . Product = 0 —> NO
 {-1, 0, 1}. Product = 0 —> NO
 {-2, -1, 0, 1, 2}. Product = 0 —> NO

II. The smallest of the k integers is zero
 {0} . Smallest = 0 —> YES
 {-1, 0, 1}. Smallest = -1 —> NO
 {-2, -1, 0, 1, 2}. Smallest = -2 —> NO

III. The largest of the k integers is negative
 {0} . Largest = 0 —> NO
 {-1, 0, 1}. Largest = 1 —> NO
 {-2, -1, 0, 1, 2}. Largest = 2 —> NO

So, I and III CANNOT be true

IMO Option D

Pls Hit kudos if you like the solution

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ScottTargetTestPrep · Answer

If the sum of k consecutive integers is 0, where k ≥ 1, then one of the numbers must be 0, and it must be the middle number if k > 1. Now, let’s analyze the Roman numeral statements.

I. The product of the k integers is positive

Since one of the integers must be 0, the product of the k integers must also be 0 . So statement I CANNOT be true.

II. The smallest of the k integers is zero

If k = 1, then the set of integers contains only the number 0, which is the smallest integer (or the largest integer) of the set. So statement II can be true.

III. The largest of the k integers is negative.

From statement II, we see that if k = 1, then the largest integer is 0. However, if k > 1, then the largest integer will be greater than 0 since 0 will be the middle integer. Therefore, no matter what, the largest integer can’t be negative. So statement III CANNOT be true.

Answer: D

Ved22 · Answer

ISNT THE QUESTION SAYING K>=1 . SO WHY NEGATIVE INTEGERS ARE CONSIDERED?

IanStewart · Answer

k is the number of integers in the list. It is not one of the integers in the list. If k = 5, for example, that means the list contains 5 consecutive integers. But some of those integers could be negative. In fact, if "the sum of k consecutive integers is 0" it must be true that some of the integers in the list are negative, unless the list contains only one value. The sum of several consecutive integers could never be zero if every one of those integers was 1 or larger.

Ankush12345 · Answer

Hey! Just wanted to ask you how can we consider a negative integer when it is given that k>=1

Bunuel · Answer

Have you misssed this post? https://gmatclub.com/forum/which-of-the ... l#p2301935

egmat · Answer

Looking at your doubt, I can see the confusion! You're mixing up what  k  represents in this problem.

Key clarification:k  is the  number of consecutive integers , NOT the value of the integers themselves.

When we say  k ≥ 1 , it means:
- We have at least  1  integer in our list
- We could have  1  integer,  3  integers,  5  integers, etc.

The actual integers can be  any values  - positive, negative, or zero!

For example:
- If  k = 3 : We have  3  consecutive integers like  -1, 0, 1 
- If  k = 5 : We have  5  consecutive integers like  -2, -1, 0, 1, 2

Notice how we include negative integers even though  k  itself is positive.

Which of the following CANNOT be true if the sum of k consecutive inte

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Which of the following CANNOT be true if the sum of k consecutive inte

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