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Difficulty: MediumConsecutive Integers and Number Sets

Set SS consists of nn consecutive integers listed in increasing order. The product of the least integer and the greatest integer in Set SS is equal to 00. If the sum of all the integers in Set SS is 2121, how many integers are in Set SS?

  1. A
    6
  2. 7Answer
  3. C
    8
  4. D
    12
  5. E
    14

Answer

7
The given sum of the consecutive integers is positive (2121), which requires the set of consecutive integers to start at 00 rather than end at 00. Setting up the sum formula for consecutive integers from 00 to kk gives k(k+1)2=21\frac{k(k+1)}{2} = 21, solving to k=6k = 6. Including 00, the terms in the set are 0,1,2,3,4,5,60, 1, 2, 3, 4, 5, 6, which yields 6−0+1=76 - 0 + 1 = 7 integers in total.

Step-by-Step Solution

1
Analyze the condition for the product of the least and greatest integers.
Since the product of the least integer aa and greatest integer bb is 00, at least one of these extreme terms must be 00.
By the zero-product property, a⋅b=0  ⟹  a=0a \cdot b = 0 \implies a = 0 or b=0b = 0.
2
Determine which extreme integer is equal to 0.
The least integer aa must be 00.
If b=0b = 0, all integers in the set would be ≤0\leq 0, so their sum would be ≤0\leq 0. Since the given sum is 21>021 > 0, the set must consist of non-negative integers starting at 00.
3
Set up the sum equation for consecutive integers starting at 0.
k(k+1)2=21  ⟹  k(k+1)=42  ⟹  k=6\frac{k(k+1)}{2} = 21 \implies k(k+1) = 42 \implies k = 6.
The sum of integers from 00 to kk is 0+k(k+1)2=210 + \frac{k(k+1)}{2} = 21.
4
Count the total number of terms in the set.
n=6−0+1=7n = 6 - 0 + 1 = 7.
The number of integers in an inclusive range from aa to bb is given by b−a+1b - a + 1.

Key Concept

Consecutive Integers, Zero Property, and Inclusive Term Counting
Estimated Time:1m 30s
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Topics

Consecutive Integers and Number Sets

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