The product of four consecutive natural numbers plus one is

Options

A. a non square
B. always sum of two square numbers
C. a square
D. None of these

Answer

To solve the problem of determining whether the product of four consecutive natural numbers plus one is a perfect square, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Four Consecutive Natural Numbers**:
Let the four consecutive natural numbers be \( n, n+1, n+2, n+3 \).

2. **Calculate the Product**:
The product of these four numbers can be expressed as:
\[ P = n \times (n+1) \times (n+2) \times (n+3) \]

3. **Add One to the Product**:
We need to find \( P + 1 \):
\[ P + 1 = n \times (n+1) \times (n+2) \times (n+3) + 1 \]

4. **Check for Perfect Square**:
We will check if \( P + 1 \) is a perfect square by substituting specific values for \( n \).

5. **Example Calculation**:
- Let’s take \( n = 2 \):
\[ P = 2 \times 3 \times 4 \times 5 = 120 \]
\[ P + 1 = 120 + 1 = 121 \]
Since \( 121 = 11^2 \), it is a perfect square.

- Now, let’s take \( n = 7 \):
\[ P = 7 \times 8 \times 9 \times 10 = 5040 \]
\[ P + 1 = 5040 + 1 = 5041 \]
Since \( 5041 = 71^2 \), it is also a perfect square.

6. **Conclusion**:
From the examples checked, we can conclude that the product of four consecutive natural numbers plus one is indeed a perfect square.