Question

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.

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 x 8 = 144

⇒ y = 12, since y is a positive integer

Hence, the two numbers are 8 and 12.