A gardener wants to plant trees in a garden. If the number of trees in each row is the same and there are 20 or 40 or 60 rows, then no tree is left. Find the least number of trees the gardener have

Options

A. 120
B. 360
C. 240
D. 200

Answer

To find the least number of trees the gardener can plant in the garden such that the number of trees in each row is the same, and there are either 20, 40, or 60 rows, we need to determine the least common multiple (LCM) of these three numbers.

### Step-by-Step Solution:

1. **Identify the Numbers**: We need to find the LCM of 20, 40, and 60.

2. **Prime Factorization**:
- For 20:
- \(20 = 2^2 \times 5^1\)
- For 40:
- \(40 = 2^3 \times 5^1\)
- For 60:
- \(60 = 2^2 \times 3^1 \times 5^1\)

3. **Determine the Highest Power of Each Prime**:
- The prime factors involved are 2, 3, and 5.
- For 2: The highest power is \(2^3\) (from 40).
- For 3: The highest power is \(3^1\) (from 60).
- For 5: The highest power is \(5^1\) (common in all).

4. **Calculate the LCM**:
- The LCM is found by multiplying the highest powers of all prime factors:
\[ LCM = 2^3 \times 3^1 \times 5^1 \]
- Calculating this:
\[ LCM = 8 \times 3 \times 5 \]
- First, calculate \(8 \times 3 = 24\).
- Then, calculate \(24 \times 5 = 120\).

5. **Conclusion**: The least number of trees the gardener can plant is **120**.