Graduate Management Admission Test (GMAT) Practice Test

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The sum of what type of integers is always a multiple of the quantity of integers involved?

Consecutive integers

The sum of consecutive integers is always a multiple of the quantity of integers involved because of the arithmetic properties of these integers.

When considering $ n $ consecutive integers, you can express them generally as $ x, x+1, x+2, \ldots, x+n-1 $. The sum of these integers can be computed as:

S = x + (x + 1) + (x + 2) + \ldots + (x + n - 1) = nx + \frac{(n - 1)n}{2}

This simplifies to:

S = nx + \frac{n(n - 1)}{2}

To see why this sum aligns with the number of integers $ n $, observe that:

- The first part, $ nx $, is clearly a multiple of $ n $.

- The second part consists of $ \frac{n(n - 1)}{2} $. For any integer $ n $, this quantity will also be divisible by $ n $ when $ n $ is greater than 1.

Thus, regardless of the specific values of $ x $ and $ n $, due to the

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Even integers

Odd integers

Prime integers

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