The LCM of two numbers is 15 times their HCF. The sum of the HCF and LCM is…

2026

The LCM of two numbers is 15 times their HCF. The sum of the HCF and LCM is 480. If both numbers are smaller than the LCM, find both the numbers.

  1. A.

    90, 150

  2. B.

    67, 250

  3. C.

    100, 520

  4. D.

    97, 200

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Show answer & explanation

Correct answer: A

For two positive integers, if H is their HCF, the numbers can be written as H × a and H × b, where a and b are coprime (they share no common factor other than 1). Their LCM is then H × a × b, so the product of the two numbers always equals HCF × LCM.

  1. Let the HCF be H. It is given that the LCM is 15 times H.

  2. It is given that HCF + LCM = 480, so H + 15H = 16H = 480, which gives H = 30.

  3. Then LCM = 15 × 30 = 450.

  4. Write the two numbers as 30a and 30b, where a and b are coprime and a × b = LCM ÷ HCF = 450 ÷ 30 = 15.

  5. The coprime factor pairs of 15 are (1, 15) and (3, 5).

  6. The pair (1, 15) gives the numbers 30 and 450 — but 450 equals the LCM itself, which violates the condition that both numbers must be smaller than the LCM. So this pair is rejected.

  7. The pair (3, 5) gives the numbers 30 × 3 = 90 and 30 × 5 = 150, both smaller than 450 — this satisfies every condition of the problem.

Check: the HCF of 90 and 150 is 30, and their LCM is 450; the sum 30 + 450 = 480 matches the given condition, and 450 = 15 × 30 confirms the LCM-to-HCF ratio.

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