Sum and LCM Numbers: Finding the Pair

The problem asks us to identify two numbers based on their sum and Least Common Multiple (L.C.M.).

Let the two numbers be $a$ and $b$. The given conditions are:

• Sum: $a + b = 132$
• L.C.M.: L.C.M.($a$, $b$) = 630

Options Verification for Sum and LCM

We can examine the provided options to find the pair that satisfies both conditions. Let's verify Option 2 (90, 42):

• Verify Sum:

  Check if the sum of 90 and 42 equals 132.

  $90 + 42 = 132$. The sum condition is satisfied.

• Verify L.C.M.:

  Calculate the L.C.M. of 90 and 42.

  First, find the prime factorization of each number:

  • $90 = 2 \times 45 = 2 \times 3 \times 15 = 2 \times 3 \times 3 \times 5 = 2 \times 3^2 \times 5$
  • $42 = 2 \times 21 = 2 \times 3 \times 7$

  The L.C.M. is found by taking the highest power of each prime factor present in either factorization:

  L.C.M.(90, 42) = $2^1 \times 3^2 \times 5^1 \times 7^1 = 2 \times 9 \times 5 \times 7 = 18 \times 35 = 630$.

  The L.C.M. condition is also satisfied.

Since the pair (90, 42) meets both the sum requirement ($132$) and the L.C.M. requirement ($630$), this is the correct pair of numbers.