To solve this problem, let's denote the three numbers by \(2x\), \(3x\), and \(4x\), where \(x\) is a scaling factor.
Given the sum of the squares of these numbers is 116, we can set up the following equation:
[tex]\[
(2x)^2 + (3x)^2 + (4x)^2 = 116
\][/tex]
Let's expand and simplify this equation:
[tex]\[
4x^2 + 9x^2 + 16x^2 = 116
\][/tex]
[tex]\[
(4 + 9 + 16)x^2 = 116
\][/tex]
[tex]\[
29x^2 = 116
\][/tex]
Next, we solve for \(x^2\) by dividing both sides of the equation by 29:
[tex]\[
x^2 = \frac{116}{29}
\][/tex]
[tex]\[
x^2 = 4
\][/tex]
Taking the square root of both sides gives us two possible values for \(x\):
[tex]\[
x = 2 \quad \text{or} \quad x = -2
\][/tex]
Since we are looking for natural numbers, we consider the positive value of \(x\):
[tex]\[
x = 2
\][/tex]
Now, we use this value of \(x\) to find the three numbers:
[tex]\[
2x = 2 \cdot 2 = 4
\][/tex]
[tex]\[
3x = 3 \cdot 2 = 6
\][/tex]
[tex]\[
4x = 4 \cdot 2 = 8
\][/tex]
Therefore, the three natural numbers are [tex]\(4\)[/tex], [tex]\(6\)[/tex], and [tex]\(8\)[/tex].
