L.C.M of `3^(3), 4 ,4^(2)` and 3 is-

Options

A. 12
B. 48
C. 432
D. None of these

Answer

To find the L.C.M. (Least Common Multiple) of the numbers \(3^3\), \(4\), \(4^2\), and \(3\), we will follow these steps:

### Step 1: Find the Prime Factorization of Each Number
1. **For \(3^3\)**:
\[ 3^3 = 27 \quad \text{(which can be expressed as } 3 \times 3 \times 3\text{)} \]
Prime factorization: \(3^3\)

2. **For \(4\)**:
\[ 4 = 2^2 \quad \text{(which can be expressed as } 2 \times 2\text{)} \]
Prime factorization: \(2^2\)

3. **For \(4^2\)**:
\[ 4^2 = 16 = 2^4 \quad \text{(which can be expressed as } 2 \times 2 \times 2 \times 2\text{)} \]
Prime factorization: \(2^4\)

4. **For \(3\)**:
\[ 3 = 3^1 \]
Prime factorization: \(3^1\)

### Step 2: Identify the Highest Power of Each Prime Factor
Now, we will take the highest power of each prime factor from the factorizations:
- For the prime \(2\): The highest power is \(2^4\) (from \(4^2\)).
- For the prime \(3\): The highest power is \(3^3\) (from \(3^3\)).

### Step 3: Calculate the L.C.M.
The L.C.M. is found by multiplying the highest powers of all prime factors together:
\[ \text{L.C.M.} = 2^4 \times 3^3 \]

Calculating each part:
- \(2^4 = 16\)
- \(3^3 = 27\)

Now, multiply these results:
\[ \text{L.C.M.} = 16 \times 27 \]

### Step 4: Perform the Multiplication
To calculate \(16 \times 27\):
\[ 16 \times 27 = 432 \]

### Final Answer
Thus, the L.C.M. of \(3^3\), \(4\), \(4^2\), and \(3\) is:
\[ \boxed{432} \]

---