Question

The sum of the squares of two positive integers is 208. If the square of the large number is 18 times the smaller. Find the numbers.

Answer 1

Let the two numbers be x and y, y being the bigger number.

From the given information,

x² + y² = 208  ...(i)

y² = 18x   ...(ii)

From (i), we get y² = 208 – x²

Putting this in (ii), we get,

208 – x² = 18x

⇒ x² + 18x – 208 = 0

⇒ x² + 26x – 8x – 208 = 0

⇒ x(x + 26) – 8(x + 26) = 0

⇒ (x – 8)(x + 26) = 0

⇒ x can't be a negative number, hence x = 8

⇒ Putting x = 8 in (ii), we get y² = 18 × 8 = 144

⇒ y = 12, since y is a positive integer

Hence, the two numbers are 8 and 12.