Graduate Management Admission Test (GMAT) Practice Test

The sum of an odd number of consecutive integers is always a multiple of:

1. The sum of the integers

2. The average of the integers

3. The number of elements in the set

4. The highest integer in the set

The correct answer is: The number of elements in the set

To understand why the correct answer is the number of elements in the set, consider the nature of consecutive integers and the properties of odd numbers. When you have an odd number of consecutive integers, you can denote them as \( n - k, n - k + 1, \ldots, n + k \), where \( n \) is the middle integer, and \( k \) represents how far from the middle the integers extend on either side. The total count of these integers is \( 2k + 1 \), which is indeed an odd number. The sum of these integers can be computed as follows: \[ \text{Sum} = (n - k) + (n - k + 1) + \ldots + (n + k) = (2n)(k + 1) = n(2k + 1) \] The total number of integers in this sequence is \( 2k + 1 \). When you look at the sum we just computed, it can be expressed as \( n \times \text{(number of integers)} \). Since the sum of the integers is thus a product of the middle number and the total count of the integers, it follows