A general can draw up his soldiers in the rows of 10, 15 and 18 soldiers and he can also draw them up in the form of a solid square. Find the least number of soldiers with the general

Options

A. 100
B. 3600
C. 900
D. 90

Answer

To solve the problem of finding the least number of soldiers the general can have, we need to follow these steps:

### Step 1: Find the LCM of the given numbers
The general can arrange his soldiers in rows of 10, 15, and 18. We need to find the least common multiple (LCM) of these numbers.

- **Prime Factorization**:
- 10 = 2 × 5
- 15 = 3 × 5
- 18 = 2 × 3²

- **Taking the highest power of each prime**:
- For 2: highest power is 2¹ (from 10 and 18)
- For 3: highest power is 3² (from 18)
- For 5: highest power is 5¹ (from 10 and 15)

- **Calculating the LCM**:
\[ \text{LCM} = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90 \]

### Step 2: Express the number of soldiers
The number of soldiers can be expressed as:
\[ \text{Number of soldiers} = 90k \]
where \( k \) is a natural number.

### Step 3: Ensure the number of soldiers forms a perfect square
To find the least number of soldiers that can also be arranged in a solid square, \( 90k \) must be a perfect square.

- **Prime factorization of 90**:
\[ 90 = 2^1 \times 3^2 \times 5^1 \]

For \( 90k \) to be a perfect square, all the exponents in its prime factorization must be even.

- **Adjusting the exponents**:
- For \( 2^1 \): we need one more 2 to make it \( 2^2 \) (even).
- For \( 3^2 \): already even.
- For \( 5^1 \): we need one more 5 to make it \( 5^2 \) (even).

Thus, we need:
\[ k = 2^1 \times 5^1 = 10 \]

### Step 4: Calculate the least number of soldiers
Now substituting \( k = 10 \) back into the equation:
\[ \text{Number of soldiers} = 90 \times 10 = 900 \]

### Conclusion
The least number of soldiers the general can have is **900**.

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