Question

The sum of the squares of three consecutive odd numbers is 2531. Find the numbers.
[Hint : Let the required odd numbers be x, x + 2 and x + 4. Then,
$$x^2 + (x + 2)^2 + (x + 4)^2 = 2531 \Rightarrow x^2 + 4x - 837 = 0 \Rightarrow x^2 + 31x - 27x - 837 = 0$$]

Answer 1

Let the three consecutive odd numbers be $$x$$, $$(x + 2)$$ and $$(x + 4)$$.

According to the problem: $$x^2 + (x + 2)^2 + (x + 4)^2 = 2531$$

$$x^2 + (x^2 + 4x + 4) + (x^2 + 8x + 16) = 2531$$

$$3x^2 + 12x + 20 = 2531$$

$$3x^2 + 12x - 2511 = 0$$

Dividing throughout by 3: $$x^2 + 4x - 837 = 0$$

Factoring: $$x^2 + 31x - 27x - 837 = 0$$

$$x(x + 31) - 27(x + 31) = 0$$

$$(x + 31)(x - 27) = 0$$

$$x = 27 \quad \text{or} \quad x = -31$$

Taking positive odd integers, $$x = 27$$.

Then, $$x + 2 = 29$$ and $$x + 4 = 31$$.

Hence, the required numbers are 27, 29 and 31.