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The LCM of two numbers is 84 and their HCF is 12. What are the two numbers? (202

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Question: The LCM of two numbers is 84 and their HCF is 12. What are the two numbers? (2023)

Options:

  1. 24 and 42
  2. 12 and 84
  3. 28 and 36
  4. 21 and 48

Correct Answer: 24 and 42

Exam Year: 2023

Solution: 

Let the two numbers be 12x and 12y. Then, LCM(12x, 12y) = 12 * LCM(x, y) = 84, which gives LCM(x, y) = 7. The pairs (x, y) that satisfy this are (3, 4) or (4, 3), leading to 24 and 42.

The LCM of two numbers is 84 and their HCF is 12. What are the two numbers? (202

Practice Questions

Q1
The LCM of two numbers is 84 and their HCF is 12. What are the two numbers? (2023)
  1. 24 and 42
  2. 12 and 84
  3. 28 and 36
  4. 21 and 48

Questions & Step-by-Step Solutions

The LCM of two numbers is 84 and their HCF is 12. What are the two numbers? (2023)
  • Step 1: Understand that the LCM (Least Common Multiple) of two numbers is 84 and their HCF (Highest Common Factor) is 12.
  • Step 2: Let the two numbers be represented as 12x and 12y, where x and y are some integers.
  • Step 3: Use the formula for LCM: LCM(12x, 12y) = 12 * LCM(x, y).
  • Step 4: Since we know LCM(12x, 12y) = 84, we can set up the equation: 12 * LCM(x, y) = 84.
  • Step 5: Divide both sides of the equation by 12 to find LCM(x, y): LCM(x, y) = 84 / 12 = 7.
  • Step 6: Now, we need to find pairs of integers (x, y) such that their LCM is 7.
  • Step 7: The pairs (x, y) that satisfy LCM(x, y) = 7 are (1, 7) and (7, 1), but we also have (3, 4) and (4, 3) since 3 and 4 are coprime.
  • Step 8: Calculate the actual numbers using the pairs: For (3, 4), the numbers are 12*3 = 36 and 12*4 = 48. For (4, 3), the numbers are 12*4 = 48 and 12*3 = 36.
  • Step 9: The two numbers are 36 and 48, or 48 and 36, which are the same numbers.
  • 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 suitable pairs that yield the correct LCM.
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