Question
The product of two positive integers is 1620. If their
HCF is 9 and their sum is 81, then find the difference between the numbers.Solution
Let the numbers be 9a and 9b
Then:
9a × 9b = 1620 ⇒ ab = 20 ...............(1)
9a + 9b = 81 ⇒ a + b = 9
Put a = (9 - b) in (1):
(9 - b) × b = 20
⇒ 9b - b² = 20
⇒ b² - 9b + 20 = 0
⇒ b² - 4b - 5b + 20 = 0
⇒ b(b - 4) - 5(b - 4) = 0
⇒ (b - 4)(b - 5) = 0
So, b = 4 or 5
Then, numbers are 36 and 45
Required difference = 9
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