The sum of the squares of two positive integers is 208. If the square of the larger number is 18 times the smaller number, find the numbers.

Let larger number be x and smaller number be y,

According to first part,

x² + y² = 208 ……..(i)

x² = 18y ……..(ii)

Substituting value of x² from (ii) in (i) we get,

⇒ 18y + y² = 208

⇒ y² + 18y - 208 = 0

⇒ y² + 26y - 8y - 208 = 0

⇒ y(y + 26) - 8(y + 26) = 0

⇒ (y - 8)(y + 26) = 0

⇒ y - 8 = 0 or y + 26 = 0

⇒ y = 8 or y = -26.

Since, numbers are positive integers,

∴ y ≠ -26.

Substituting value of y = 8 in (ii),

⇒ x² = 18(8) = 144

⇒ x = $\sqrt{144} = \pm 12$.

Since, numbers are positive integers,

∴ x ≠ -12.

Hence, numbers are 12 and 8.