Each student of class 10 contributed some money for a picnic. The money contributed by each student was equal to the cube of the total number of students. If the total collected amount was Rs29791, then find the total number of students.

Options

A. 15
B. 27
C. 31
D. 34

Answer

To solve the problem step by step, we will follow these procedures:

### Step 1: Define the Variables
Let the total number of students be denoted as \( x \).

### Step 2: Express the Contribution
According to the problem, the money contributed by each student is equal to the cube of the total number of students. Therefore, the total contribution from all students can be expressed as:
\[ \text{Total Amount} = x \cdot (x^3) = x^4 \]
This means that the total amount collected is \( x^4 \).

### Step 3: Set Up the Equation
We know from the problem statement that the total collected amount is Rs 29,791. Thus, we can set up the equation:
\[ x^4 = 29791 \]

### Step 4: Solve for \( x \)
To find \( x \), we need to take the fourth root of both sides:
\[ x = \sqrt[4]{29791} \]

### Step 5: Factor 29791
To simplify \( \sqrt[4]{29791} \), we can factor 29791. We can start by checking divisibility by smaller prime numbers.

1. **Dividing by 31**:
\[ 29791 \div 31 = 961 \]
2. **Dividing 961 by 31**:
\[ 961 \div 31 = 31 \]
3. **Dividing 31 by 31**:
\[ 31 \div 31 = 1 \]

So, we have:
\[ 29791 = 31^3 \]

### Step 6: Calculate the Fourth Root
Now we can express \( 29791 \) as:
\[ 29791 = (31^3) \]
To find \( x \):
\[ x = \sqrt[4]{31^3} = 31^{3/4} \]
This means:
\[ x = 31 \]

### Conclusion
Thus, the total number of students is:
\[ \boxed{31} \]