Question

The difference of the squares of two positive integers is 180. The square of the smaller number is 8 times the larger, find the numbers.

Answer 1

Let the larger number be x.

Then according to the question,

Square of the smaller number = 8x, then

$$\begin{gathered} x^2-8x=180 \\ \Rightarrow x^2-8x-180=0 \\ \Rightarrow x^2-18x+10x-180=0 \\ \Rightarrow x(x-18)+10(x-18)=0 \\ \Rightarrow (x+10)(x-18)=0 \\ \Rightarrow x+10=0\mathrm{or}x-18=0 \\ \Rightarrow x=-10\mathrm{or}x=18 \end{gathered}$$

Since, x being a positive integer so, x cannot be negative,

Therefore, larger number = 18.

then the smaller number = $\sqrt{8\times 18}=12$

Thus, the two positive numbers are 12 and 18.