If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is
- 1777
- 1785
- 1875
- 1877
The Correct Option is D
Solution and Explanation
To find the sum of the squares of three consecutive positive integers whose product is 15600, follow these steps:
Let the three consecutive integers be \(x-1\), \(x\), and \(x+1\).
Their product is given by \((x-1)x(x+1)\). We set up the equation:
\((x-1)x(x+1) = 15600\)
This simplifies to:
\(x(x^2-1) = 15600\)
\(x^3 - x = 15600\)
Estimate \(x\) by trying values close to the cube root of 15600. We find \(x \approx 25\).
Verify by calculating the product:
If \(x = 25\), the numbers are \(24\), \(25\), \(26\).
\(24 \times 25 \times 26 = 15600\)
The estimation is correct.
Calculate the sum of squares:
\((24)^2 + (25)^2 + (26)^2\)
= \(576 + 625 + 676\)
= \(1877\)
Therefore, the sum of the squares of these integers is 1877.
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