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Difficulty: HardConsecutive Integers and Number Sets

Set SS consists of nn consecutive integers. The sum of all elements in set SS is equal to 00. If the product of the smallest element in set SS and the total number of elements nn is −300-300, how many positive integers are in set SS?

  1. A
    11
  2. B
    13
  3. 12Answer
  4. D
    24
  5. E
    25

Answer

12
For a set of consecutive integers to sum to zero, the elements must be symmetric about zero. Therefore, the set has an odd number of terms n=2k+1n = 2k + 1 ranging from −k-k to kk. Setting up the product of the smallest term (−k)(-k) and the term count (2k+1)(2k + 1) gives (−k)(2k+1)=−300(-k)(2k + 1) = -300, which simplifies to 2k2+k−300=02k^2 + k - 300 = 0. Factoring this quadratic yields (2k+25)(k−12)=0(2k + 25)(k - 12) = 0, so k=12k = 12. The positive integers in the set are 11 through 1212, totaling 12 elements.

Step-by-Step Solution

1
Determine the symmetry and structure of set SS using its sum.
The median of set SS is 00, and the set contains an odd number of terms n=2k+1n = 2k + 1 centered at 00, expressed as {−k,−(k−1),…,0,…,k−1,k}\{-k, -(k-1), \dots, 0, \dots, k-1, k\}.
For a set of consecutive integers to sum to 00, the terms must be symmetric around 00. Since 00 is an element of the set, nn must be odd.
2
Set up the quadratic equation using the given product.
The smallest element is −k-k and the total number of elements is n=2k+1n = 2k + 1, yielding (−k)(2k+1)=−300(-k)(2k + 1) = -300, which simplifies to 2k2+k−300=02k^2 + k - 300 = 0.
The question specifies that the product of the smallest element and the number of elements is −300-300.
3
Solve the quadratic equation for kk.
Factoring (2k+25)(k−12)=0(2k + 25)(k - 12) = 0 gives k=12k = 12 as the only positive integer solution.
Since kk represents a count of elements strictly above zero, kk must be a positive integer.
4
Count the number of positive integers in set SS.
The positive integers are 1,2,…,121, 2, \dots, 12, which gives a total of 1212 positive integers.
Zero is neither positive nor negative, so only the integers from 11 to kk (1212) are counted.

Key Concept

Symmetry and median property of consecutive integers centered at zero.
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Topics

Consecutive Integers and Number Sets

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