Find the LCM of the numbers given below :
36, 40, 126

Options

A. `2410`
B. `2420`
C. `2510`
D. `2520`

Answer

To find the LCM (Least Common Multiple) of the numbers 36, 40, and 126, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **For 36:**
- 36 can be divided by 2: \( 36 \div 2 = 18 \)
- 18 can be divided by 2: \( 18 \div 2 = 9 \)
- 9 can be divided by 3: \( 9 \div 3 = 3 \)
- 3 can be divided by 3: \( 3 \div 3 = 1 \)

So, the prime factorization of 36 is:
\[ 36 = 2^2 \times 3^2 \]

- **For 40:**
- 40 can be divided by 2: \( 40 \div 2 = 20 \)
- 20 can be divided by 2: \( 20 \div 2 = 10 \)
- 10 can be divided by 2: \( 10 \div 2 = 5 \)
- 5 can be divided by 5: \( 5 \div 5 = 1 \)

So, the prime factorization of 40 is:
\[ 40 = 2^3 \times 5^1 \]

- **For 126:**
- 126 can be divided by 2: \( 126 \div 2 = 63 \)
- 63 can be divided by 3: \( 63 \div 3 = 21 \)
- 21 can be divided by 3: \( 21 \div 3 = 7 \)
- 7 can be divided by 7: \( 7 \div 7 = 1 \)

So, the prime factorization of 126 is:
\[ 126 = 2^1 \times 3^2 \times 7^1 \]

### Step 2: Identify the Highest Powers of Each Prime Factor
Now, we will take the highest power of each prime factor from the factorizations:

- For \(2\): The highest power is \(2^3\) (from 40).
- For \(3\): The highest power is \(3^2\) (from both 36 and 126).
- For \(5\): The highest power is \(5^1\) (from 40).
- For \(7\): The highest power is \(7^1\) (from 126).

### Step 3: Multiply the Highest Powers Together
Now, we multiply these highest powers together to find the LCM:

\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 \times 7^1 \]

Calculating this step-by-step:

1. \(2^3 = 8\)
2. \(3^2 = 9\)
3. \(5^1 = 5\)
4. \(7^1 = 7\)

Now multiply them together:

\[ 8 \times 9 = 72 \]
\[ 72 \times 5 = 360 \]
\[ 360 \times 7 = 2520 \]

Thus, the LCM of 36, 40, and 126 is:
\[ \text{LCM} = 2520 \]

### Final Answer
The LCM of the numbers 36, 40, and 126 is **2520**.

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