The H.C.F. of two numbers is 18. First number is of three digits and Second…

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

Step-by-Step Calculation

1. Define the numbers: Since the H.C.F. is 18, we can represent the two numbers as:

• First number = 18x

• Second number = 18y (where x and y are co-prime, meaning they share no common factors).

2. Relate to the L.C.M.: We know that the L.C.M. is 18xy.

• 18xy = 936

• xy = 936 / 18

• xy = 52

3. Find co-prime pairs (x, y) that multiply to 52: The pairs of factors for 52 are:

• (1, 52)

• (2, 26) - Invalid (not co-prime, they share a factor of 2)

• (4, 13)

4. Determine the numbers based on digit constraints:

• Pair (1, 52):

  • 18 × 1 = 18 (2 digits)

  • 18 × 52 = 936 (3 digits)

  • This pair satisfies the condition (one is 3 digits, one is 2 digits).

• Pair (4, 13):

  • 18 × 4 = 72 (2 digits)

  • 18 × 13 = 234 (3 digits)

  • This pair also satisfies the condition.

Looking at the options provided (276, 306, 278, 310), let's check the sums:

• Sum of pair (1, 52) = 18 + 936 = 954

• Sum of pair (4, 13) = 72 + 234 = 306

The sum 306 corresponds to Option 2.