Question
The product of two positive integers is 2646. If their
HCF is 7 and their sum is 105, then find the difference between the numbers.Solution
Let the numbers be 7a and 7b
Then:
7a × 7b = 2646 ⇒ ab = 54 ...............(1)
7a + 7b = 105 ⇒ a + b = 15
Put a = (15 - b) in (1):
(15 - b) × b = 54
⇒ 15b - b² = 54
⇒ b² - 15b + 54 = 0
⇒ b² - 6b - 9b + 54 = 0
⇒ b(b - 6) - 9(b - 6) = 0
⇒ (b - 6)(b - 9) = 0
So, b = 6 or 9
Then, numbers are 42 and 63
Required difference = 21
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