The minimum soldiers that can be arranged in 12, 15 and 18 rows of equal soldiers, and also arrange them in squarical way minimum No. will be -

Options

A. 180
B. 450
C. 900
D. 32400

Answer

To find the minimum number of soldiers that can be arranged in 12, 15, and 18 rows of equal soldiers, and also arranged in a square formation, we need to calculate the Least Common Multiple (LCM) of the numbers 12, 15, and 18. Here’s how we can do it step by step:

### Step 1: Find the Prime Factorization of Each Number
- **12**: The prime factorization of 12 is \(2^2 \times 3^1\).
- **15**: The prime factorization of 15 is \(3^1 \times 5^1\).
- **18**: The prime factorization of 18 is \(2^1 \times 3^2\).

### Step 2: Identify the Highest Power of Each Prime Factor
Next, we identify the highest power of each prime factor that appears in the factorizations:
- For the prime number **2**, the highest power is \(2^2\) (from 12).
- For the prime number **3**, the highest power is \(3^2\) (from 18).
- For the prime number **5**, the highest power is \(5^1\) (from 15).

### Step 3: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \]

Calculating this step by step:
1. Calculate \(2^2 = 4\).
2. Calculate \(3^2 = 9\).
3. Calculate \(5^1 = 5\).
4. Now multiply these results together:
\[ 4 \times 9 = 36 \]
\[ 36 \times 5 = 180 \]

### Final Answer
Thus, the minimum number of soldiers that can be arranged in 12, 15, and 18 rows of equal soldiers, and also arranged in a square formation, is **180**.