Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is a perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say m, of these three integers?
Explanation:
Let the three numbers be (a – 2), (a – 1) and a.
∴ (a – 2) + (a – 1)2 + a3 = p2
Where p is the sum of the three integers.
Now, a – 2 + a2 – 2a + 1 + a3 = p2
∴ a3 + a2 – a – 1 = p2
∴ a2(a + 1) – 1(a + 1) = p2
∴ (a2 – 1)(a + 1) = p2
∴ (a + 1)2(a – 1) = p2
For the above condition to be satisfied, (a – 1) must be a perfect square.
The smallest possible value for a - 1 is 4, since (a – 2) cannot be zero, giving us (a − 2) = 3
The minimum of the three is therefore 3.
Hence, option (a).
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