Question

If the LCM of 12 and 42 is 10m + 4, then the value of m is

• A. \(\frac{1}{5}\)
• B. \(\frac{4}{5}\)
• C. 5
• D. 8

Answer 1

The Correct Option is D

To solve the problem, we need to find the value of \( m \) in the equation of the form \( \text{LCM}(12, 42) = 10m + 4 \). 
The problem provides options and the solution is 8, but let's understand how to derive it:

Step 1: Calculate the LCM of 12 and 42.
To find the LCM, we first determine the prime factorizations:
12 = \( 2^2 \times 3 \)
42 = \( 2 \times 3 \times 7 \)
The LCM is given by taking the highest power of each prime number appearing in the factorizations:
LCM = \( 2^2 \times 3 \times 7 = 4 \times 3 \times 7 \)
\( = 84 \)

Step 2: Use the LCM in the equation provided in the problem.
We have \( 84 = 10m + 4 \).
Rearrange to solve for \( m \):
\( 84 - 4 = 10m \)
\( 80 = 10m \)
Divide both sides by 10 to isolate \( m \):
\( m = \frac{80}{10} = 8 \)
Hence, the value of \( m \) is 8.

Answer 2

To find the value of \( m \), we first determine the LCM of 12 and 42.

Step 1: Prime Factorization

• \( 12 = 2^2 \times 3 \)
• \( 42 = 2 \times 3 \times 7 \) 

Step 2: Compute LCM

The LCM is found by taking the highest power of each prime:

\( \text{LCM} = 2^2 \times 3 \times 7 = 84 \)

Step 3: Solve for \( m \)

Given \( 10m + 4 = 84 \), solving for \( m \):

\( 10m = 84 - 4 \)

\( 10m = 80 \)

\( m = \frac{80}{10} = 8 \)

Final Answer: 8