Question

Three consecutive positive integers are raised to the first, second, and third powers respectively and added. The sum is a perfect square whose square root equals the total of the three original integers. Which range best describes the minimum integer \(m\) of these three?

• A. \(1 \le m \le 3\)
• B. \(4 \le m \le 6\)
• C. \(7 \le m \le 9\)
• D. \(10 \le m \le 12\)
• E. \(13 \le m \le 15\)

Hint:

Always test small ranges after setting up the algebraic condition to quickly find the valid range.

Answer 1

The Correct Option is C

Let integers be \(m, m+1, m+2\). Sum: \(m^1 + (m+1)^2 + (m+2)^3\). Condition: \(\sqrt{\text{Sum}} = 3m + 3\). Squaring and solving shows smallest \(m\) satisfying is in range \(7 \le m \le 9\).