Question
The product of two positive integers is 1400. If their HCF
is 5 and their sum is 75, then find the difference between the numbers.Solution
Let the numbers be 5a and 5b
Then:
5a × 5b = 1400 ⇒ ab = 56 ...............(1)
5a + 5b = 75 ⇒ a + b = 15
Put a = (15 - b) in (1):
(15 - b) × b = 56
⇒ 15b - b² = 56
⇒ b² - 15b + 56 = 0
⇒ b² - 7b - 8b + 56 = 0
⇒ b(b - 7) - 8(b - 7) = 0
⇒ (b - 7)(b - 8) = 0
So, b = 8 or 7
Then, numbers are 40 and 35
Required difference = 5
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