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CAT 2017 Slot 2 — QA Question 21

Number TheoryEasy

If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is

Answer & solution

Correct answer: 1877

  • A

    1777

  • B

    1785

  • C

    1875

  • 1877

Solution

Easy

Three consecutive integers multiply to roughly the cube of the middle one. Cube-root 1560015600 to estimate the middle integer, confirm by factoring, then add the squares.

1

Estimate the middle integer. The product of three consecutive integers is close to the cube of the middle one, and 253=15625≈1560025^3 = 15625 \approx 15600.

156003≈25(since 253=15625)\begin{aligned} &\sqrt[3]{15600} \approx 25 \quad\text{(since } 25^3 = 15625\text{)} \end{aligned}
2

Confirm the trio. Divide out the middle integer and factor the rest.

1560025=624=24×26(neighbours of 25)⇒ 24×25×26=15600✓\begin{aligned} &\frac{15600}{25} = 624 = 24\times 26 \quad\text{(neighbours of 25)}\\ &\Rightarrow\ 24\times 25\times 26 = 15600 \checkmark \end{aligned}
3

Sum of squares. Add the squares of 24,25,2624,25,26.

242+252+262=576+625+676⇒ =1877\begin{aligned} &24^2 + 25^2 + 26^2 = 576 + 625 + 676\\ &\Rightarrow\ = 1877 \end{aligned}
242+252+262=187724^2 + 25^2 + 26^2 = 1877

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CAT 2017 Slot 2 QA Q21: If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers i — Solution | TheCATExam