The sum of a natural number and its square equals the product of the first three prime numbers. The number is

Options

A. 2
B. 3
C. 5
D. 6

Answer

To solve the problem step by step, we need to find a natural number \( X \) such that the sum of the number and its square equals the product of the first three prime numbers.

### Step 1: Identify the first three prime numbers
The first three prime numbers are:
- 2 (first prime)
- 3 (second prime)
- 5 (third prime)

### Step 2: Calculate the product of the first three prime numbers
The product of these prime numbers can be calculated as:
\[ 2 \times 3 \times 5 = 30 \]

### Step 3: Set up the equation
According to the problem, the sum of the natural number \( X \) and its square \( X^2 \) equals the product we just calculated:
\[ X + X^2 = 30 \]

### Step 4: Rearrange the equation
Rearranging the equation gives us:
\[ X^2 + X - 30 = 0 \]

### Step 5: Factor the quadratic equation
To factor the quadratic equation \( X^2 + X - 30 = 0 \), we need to find two numbers that multiply to \(-30\) and add to \(1\). The numbers \(6\) and \(-5\) work:
\[ (X + 6)(X - 5) = 0 \]

### Step 6: Solve for \( X \)
Setting each factor equal to zero gives us:
1. \( X + 6 = 0 \) → \( X = -6 \) (not a natural number)
2. \( X - 5 = 0 \) → \( X = 5 \) (valid natural number)

### Conclusion
The natural number \( X \) is:
\[ \boxed{5} \]