The difference of the squares of two positive integers is 180. The squ

1.	The difference of the squares of two positive integers is 180. The square of the smaller number is 8 times the larger number, find the numbers.
Answer» Let the positive integers be ‘a’ and ‘b’. Given, difference of the squares of two positive integers is 180. ⇒ a² – b² = 180 Also, square of the smaller number is 8 times the larger. ⇒ b² = 8a Thus, a² – 8a – 180 = 0 ⇒ a² – 18a + 10a – 180 = 0 ⇒ a(a – 18) + 10(a – 18) = 0 ⇒ (a + 10)(a – 18) = 0 ⇒ a = -10, 18 Thus, the other number is 324 – 180 = b² ⇒ b = 12 Numbers are 12, 18

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

Answer»

Let the positive integers be ‘a’ and ‘b’.

Given, difference of the squares of two positive integers is 180.

⇒ a² – b² = 180

Also, square of the smaller number is 8 times the larger.

⇒ b² = 8a

Thus, a² – 8a – 180 = 0

⇒ a² – 18a + 10a – 180 = 0

⇒ a(a – 18) + 10(a – 18) = 0

⇒ (a + 10)(a – 18) = 0

⇒ a = -10, 18

Thus, the other number is 324 – 180 = b²

⇒ b = 12

Numbers are 12, 18