2026 AMC 8 Problem 18: Solution

Work through 2026 AMC 8 Problem 18 on number representation with a written solution, hints, and a visual explanation.

Number TheoryLevel 4 · Challenge

In how many ways can 60 be written as the sum of two or more consecutive odd positive integers that are arranged in increasing order?

  • A. 1
  • B. 2
  • C. 3
  • D. 4
  • E. 5
Hints
  1. Write the sum of n consecutive odd integers starting at a in terms of n and a.
  2. Set that expression equal to 60, then think about what parity n must have, given that a itself is odd.
  3. Test the possible even values of n one at a time to see which give a positive odd value for a.
Read the step-by-step solution

Answer: B · 2

  1. Express the sum algebraically

    The sum of n consecutive odd integers starting from a is a + (a+2) + ⋯ + (a+2(n−1)) = an + n(n−1) = n(a+n−1). So n(a+n−1) = 60.

    n(a + n − 1) = 60

  2. Find valid solutions

    We need n ≥ 2, a odd and positive. Since a is odd and 60 is even, n must be even. Testing: n=2 gives a=29 (valid); n=6 gives a=5 (valid). No other even n works.

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