A man plants his orchard with `5625` trees and arranges them so that there are as many rows as there are trees in each row. How many rows are there ?

Options

A. `78`
B. `65`
C. `78`
D. `75`

Answer

To solve the problem of how many rows a man can arrange his 5625 trees such that the number of rows is equal to the number of trees in each row, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We need to find a number \( X \) such that the total number of trees (5625) can be arranged in \( X \) rows, with \( X \) trees in each row.
- This means the total number of trees can be expressed as \( X \times X = X^2 \).

2. **Setting Up the Equation**:
- From the above understanding, we can set up the equation:
\[ X^2 = 5625 \]

3. **Finding the Square Root**:
- To find \( X \), we need to calculate the square root of 5625:
\[ X = \sqrt{5625} \]

4. **Using Factorization to Find the Square Root**:
- We can find the square root of 5625 by factorizing it:
- Start dividing by the smallest prime number:
- \( 5625 \div 5 = 1125 \)
- \( 1125 \div 5 = 225 \)
- \( 225 \div 5 = 45 \)
- \( 45 \div 5 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- So, the prime factorization of 5625 is:
\[ 5625 = 5^4 \times 3^2 \]

5. **Pairing the Factors**:
- To find the square root, we pair the factors:
- From \( 5^4 \), we can take out \( 5^2 \) (which is 25).
- From \( 3^2 \), we can take out \( 3 \).
- Therefore, the square root is:
\[ \sqrt{5625} = 5^2 \times 3 = 25 \times 3 = 75 \]

6. **Conclusion**:
- The number of rows \( X \) is 75.
- Therefore, there are 75 rows, and each row contains 75 trees.

### Final Answer:
The number of rows is **75**.