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If the LCM of two numbers is 60 and their HCF is 5, what is the sum of the two n

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Question: If the LCM of two numbers is 60 and their HCF is 5, what is the sum of the two numbers if they are both less than 30? (2023)

Options:

  1. 25
  2. 35
  3. 40
  4. 30

Correct Answer: 30

Exam Year: 2023

Solution: 

Let the two numbers be 5a and 5b. Then, LCM(5a, 5b) = 5 * LCM(a, b) = 60, which gives LCM(a, b) = 12. The pairs (3, 4) work, giving numbers 15 and 20, which sum to 35.

If the LCM of two numbers is 60 and their HCF is 5, what is the sum of the two n

Practice Questions

Q1
If the LCM of two numbers is 60 and their HCF is 5, what is the sum of the two numbers if they are both less than 30? (2023)
  1. 25
  2. 35
  3. 40
  4. 30

Questions & Step-by-Step Solutions

If the LCM of two numbers is 60 and their HCF is 5, what is the sum of the two numbers if they are both less than 30? (2023)
  • Step 1: Understand that the LCM (Least Common Multiple) of two numbers is 60 and their HCF (Highest Common Factor) is 5.
  • Step 2: Let the two numbers be represented as 5a and 5b, where 'a' and 'b' are integers.
  • Step 3: Use the relationship between LCM and HCF: LCM(5a, 5b) = (5a * 5b) / HCF(5a, 5b).
  • Step 4: Substitute the known values: LCM(5a, 5b) = 60 and HCF(5a, 5b) = 5.
  • Step 5: This gives us the equation: 60 = (5a * 5b) / 5, which simplifies to 60 = 5ab.
  • Step 6: Rearranging gives us ab = 12.
  • Step 7: Now, we need to find pairs of integers (a, b) such that their product is 12.
  • Step 8: The pairs (a, b) that work are (3, 4) and (4, 3).
  • Step 9: Calculate the actual numbers: If (a, b) = (3, 4), then the numbers are 5*3 = 15 and 5*4 = 20.
  • Step 10: Check if both numbers are less than 30: Yes, both 15 and 20 are less than 30.
  • Step 11: Find the sum of the two numbers: 15 + 20 = 35.
  • LCM and HCF Relationship – Understanding the relationship between the Least Common Multiple (LCM) and Highest Common Factor (HCF) of two numbers, specifically that the product of the two numbers equals the product of their LCM and HCF.
  • Factorization – Using factorization to express the numbers in terms of their HCF and finding pairs that satisfy the LCM condition.
  • Sum of Numbers – Calculating the sum of two numbers after determining their values based on given conditions.
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