CAT 2008 — QA Question 17
Directions for next 2 questions:
The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.
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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?
Answer & solution
1 ≤ m ≤ 3
- B
4 ≤ m ≤ 6
- C
7 ≤ m ≤ 9
- D
10 ≤ m ≤ 12
- E
13 ≤ m ≤ 15
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).