In a garden, there are 75 mango trees, 90 lemon and 120 orange trees respectively. The no. of trees is equal in each row and the variety is also same. Find the minimum no of rows.

Options

A. 19
B. 15
C. 29
D. None of these

Answer

To find the minimum number of rows in which the trees can be arranged such that each row has an equal number of trees of the same variety, we need to determine the highest common factor (HCF) of the number of each type of tree. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the Number of Trees**:
- Mango trees = 75
- Lemon trees = 90
- Orange trees = 120

2. **Find the Prime Factorization of Each Number**:
- For 75:
- 75 = 3 × 5 × 5 = 3 × 5²
- For 90:
- 90 = 2 × 3 × 5 × 3 = 2 × 3² × 5
- For 120:
- 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5

3. **List the Prime Factors**:
- 75: 3, 5, 5
- 90: 2, 3, 3, 5
- 120: 2, 2, 2, 3, 5

4. **Identify the Common Factors**:
- The common prime factors among 75, 90, and 120 are 3 and 5.

5. **Determine the HCF**:
- The HCF is found by taking the lowest power of all common prime factors:
- For 3: minimum power is 1 (from all three numbers)
- For 5: minimum power is 1 (from all three numbers)
- Therefore, HCF = 3¹ × 5¹ = 3 × 5 = 15.

6. **Calculate the Minimum Number of Rows**:
- To find the number of rows, divide the number of each type of tree by the HCF:
- Mango trees: 75 ÷ 15 = 5 rows
- Lemon trees: 90 ÷ 15 = 6 rows
- Orange trees: 120 ÷ 15 = 8 rows

7. **Conclusion**:
- The minimum number of rows will be determined by the type of tree that requires the most rows, which is the orange trees. Therefore, the minimum number of rows needed is **8**.

### Final Answer:
The minimum number of rows is **8**.