Two number, both greater than 27, have H.C.F 27 and L.C.M 162. The sum of the…

To find the sum of two numbers given their Highest Common Factor (H.C.F.) and Least Common Multiple (L.C.M.), we can use the following relationship:

Step-by-Step Calculation

1. Understand the Relationship: Any two numbers with an H.C.F. of H can be represented as H × a and H × b, where a and b are co-prime numbers (they have no common factors).

• The L.C.M. is then given by H × a × b.

• The product of the two numbers is H² × a × b.

2. Apply the given values:

• H.C.F. (H) = 27

• L.C.M. = 162

Find the product of the co-prime factors (a × b):

• L.C.M. = H × a × b

• 162 = 27 × (a × b)

• a × b = 162 / 27 = 6

3. Find the co-prime pairs (a, b) that multiply to 6: The possible pairs for (a, b) are:

• (1, 6)

• (2, 3)

4. Determine the actual numbers: Since the numbers must be greater than 27:

• Pair (1, 6):

  • Number 1 = 27 × 1 = 27 (Not greater than 27)

  • Number 2 = 27 × 6 = 162

  • (This pair is invalid because one number is not greater than 27)

• Pair (2, 3):

  • Number 1 = 27 × 2 = 54 (Greater than 27)

  • Number 2 = 27 × 3 = 81 (Greater than 27)

  • (This pair is valid)

5. Calculate the sum:

• Sum = 54 + 81 = 135.