Question

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

• A. 1777
• B. 1785
• C. 1875
• D. 1877

Answer 1

The Correct Option is D

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.