A man, after a tour, finds that he had spent every day as many rupees as the number of days he had been on tour. How long did his tour last, if he had spent in all `Rs. 1,296`?

Answer

To solve the problem step by step, we need to find out how long the man's tour lasted, given that he spent a certain amount of money each day equal to the number of days he was on tour, and the total amount spent was Rs. 1,296.

### Step 1: Define the Variables
Let the number of days the man was on tour be \( x \). According to the problem, he spent \( x \) rupees each day.

### Step 2: Set Up the Equation
Since he spent \( x \) rupees each day for \( x \) days, the total amount spent can be expressed as:
\[ \text{Total Amount} = \text{Number of Days} \times \text{Amount Spent per Day} \]
Thus, we have:
\[ x \times x = 1296 \]
This simplifies to:
\[ x^2 = 1296 \]

### Step 3: Solve for \( x \)
To find \( x \), we need to take the square root of both sides of the equation:
\[ x = \sqrt{1296} \]

### Step 4: Calculate the Square Root
Now, we need to calculate \( \sqrt{1296} \). We can do this through prime factorization:
1. Start dividing 1296 by the smallest prime number, which is 2:
- \( 1296 \div 2 = 648 \)
- \( 648 \div 2 = 324 \)
- \( 324 \div 2 = 162 \)
- \( 162 \div 2 = 81 \)
2. Now, 81 is not divisible by 2, so we move to the next prime number, which is 3:
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

So, the prime factorization of 1296 is:
\[ 1296 = 2^4 \times 3^4 \]

### Step 5: Apply the Square Root to the Prime Factorization
Now we can find the square root:
\[ \sqrt{1296} = \sqrt{2^4 \times 3^4} = \sqrt{(2^2)^2 \times (3^2)^2} = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 6: Conclusion
Thus, the number of days the man was on tour is:
\[ \boxed{36} \]