Set S consists of n consecutive integers. The product of the smallest and largest integers… — GMAT

Set $S$ consists of $n$ consecutive integers. The product of the smallest and largest integers in $S$ is $-144$ . If the sum of all elements in $S$ is $91$ , what is the value of $n$ ?

Answer: 26

Answer

The number of consecutive integers

n

in Set

S

is 26.

By combining the consecutive set sum formula

n(a + b) = 182

with the term span relation

b - a = n - 1

and the product constraint

ab = -144

, testing the factors of

182

uniquely determines

n = 26

, corresponding to the set of consecutive integers from

-9

16

Step-by-Step Solution

Relate the number of elements

n

to the smallest element

a

and largest element

b

b - a = n - 1

In any set of

n

consecutive integers, the distance between the maximum and minimum elements is

n - 1

Use the sum formula for an evenly spaced set.

n(a + b) = 182

The sum is

\text{Sum} = n \times \frac{a + b}{2} = 91

, which simplifies to

n(a + b) = 182

Factor

182

to find integer solutions satisfying

a \cdot b = -144

n = 26

a = -9

b = 16

For

n = 26

a + b = 7

and

b - a = 25

, giving

a = -9

and

b = 16

. Their product is

(-9)(16) = -144

Key Concept

Properties of consecutive integer sets, average/median sum formula, and term boundary indexing

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