Graduate Management Admission Test (GMAT) Practice Test

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

1. The lowest integer in the set

2. The number of elements in the set

3. The largest integer in the set

4. The average of the integers

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

The sum of an odd number of consecutive integers can be demonstrated to always be a multiple of the number of elements in that set due to the nature of how integers and their properties work. When you sum an odd number of consecutive integers, the total can be expressed as follows: Let’s say the first integer in the series is denoted as \( n \). If there are \( k \) consecutive integers (where \( k \) is odd), the integers will span from \( n \) to \( n + (k - 1) \). The sum of these integers can be calculated using the formula for the sum of an arithmetic series, which, in this case, simplifies to \( k \times \frac{(n + (n + k - 1))}{2} \). Further simplification shows that this sum can also be expressed as: \[ \text{Sum} = \text{Average} \times k \] Given that \( k \) is the total number of integers, it is multiplied with the average to yield the sum. Since \( k \) is odd, the result of this sum must inherently be a multiple of \( k \) because the product of any integer with \( k