Mr. Baghwan wants to plant 36 mango trees, 144 orange trees and 234 apple trees in his garden. If he wants to plant the equal no. of trees in every row, but the rows of mango , oranges and apple tress will be separate, then the minimum number of rows in his garden is:

Options

A. 18
B. 23
C. 36
D. can't be determined

Answer

To find the minimum number of rows in which Mr. Baghwan can plant his trees, we need to determine how many trees he can plant in each row for mango, orange, and apple trees separately. This requires finding the highest common factor (HCF) of the number of trees he has for each type.

### Step 1: Identify the number of trees
- Mango trees: 36
- Orange trees: 144
- Apple trees: 234

### Step 2: Find the prime factorization of each number
- **Mango trees (36)**:
- 36 = 2 × 2 × 3 × 3 = \(2^2 \times 3^2\)

- **Orange trees (144)**:
- 144 = 2 × 2 × 2 × 2 × 3 × 3 = \(2^4 \times 3^2\)

- **Apple trees (234)**:
- 234 = 2 × 3 × 3 × 13 = \(2^1 \times 3^2 \times 13^1\)

### Step 3: Find the HCF of the three numbers
To find the HCF, we take the lowest power of each prime factor that appears in all three factorizations:
- For 2: The lowest power is \(2^1\) (from 234).
- For 3: The lowest power is \(3^2\) (common in all).
- For 13: It does not appear in all three, so we do not include it.

Thus, the HCF is:
\[ HCF = 2^1 \times 3^2 = 2 \times 9 = 18 \]

### Step 4: Calculate the number of rows for each type of tree
Now, we can calculate how many rows will be needed for each type of tree by dividing the total number of trees by the HCF:
- **Mango trees**: \( \frac{36}{18} = 2 \) rows
- **Orange trees**: \( \frac{144}{18} = 8 \) rows
- **Apple trees**: \( \frac{234}{18} = 13 \) rows

### Step 5: Calculate the total number of rows
Now, we add the number of rows for each type of tree:
\[ \text{Total rows} = 2 + 8 + 13 = 23 \]

### Final Answer
The minimum number of rows in Mr. Baghwan's garden is **23**.

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