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.

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12 and 18.

15 and 18.

16 and 18.

17 and 18.

answer is A.

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Detailed Solution

Given that the difference of the squares of two positive integers is 180.
The square of the smaller number is 8 times the larger number.
Consider the larger number = x.
Then the square of the smaller number = 8x
According to the given condition,

x^{2} - 8 x = 180

⇒

x^{2} - 8 x - 180 = 0

⇒

x^{2} - 18 x + 10 x - 180 = 0

⇒

x (x - 18) + 10 (x - 18) = 0

⇒

x - 18 = 0 or x + 10 = 0

⇒

x = 18, - 10

Thus, the larger number = 18 or -10.
Then, the square of the smaller number is 8(18) or 8(-10)
⇒ 144 or -80
Analyze that the square of a number can't be negative, so, the square of smaller number = 144
Hence, the smaller number =