Given n = 1 + x and x is the product of four
consecutive integers. Then which of the following
is true?

I. n is an odd integer.

II. n is prime.

III. n is a perfect square.

Question

Given n = 1 + x and x is the product of four
consecutive integers. Then which of the following
is true?

I. n is an odd integer.

II. n is prime.

III. n is a perfect square.

Accepted Answer

We are given that $$ n = 1 + x $$, where $$ x $$ is the product of four consecutive integers. We need to analyze the properties of $$ n $$ based on three statements:  
1. $$ n $$ is an odd integer.  
2. $$ n $$ is prime.  
3. $$ n $$ is a perfect square.

---

### Step 1: Analyze the Parity of $$ n $$  
The product of four consecutive integers ($$ x $$) always includes at least two even numbers, making $$ x $$ an even number. Adding 1 to an even number results in an odd number.  
Thus, **$$ n $$ is always odd**.

---

### Step 2: Check if $$ n $$ is Prime  
To determine if $$ n $$ is prime, consider specific examples:  
- For the integers $$ 1, 2, 3, 4 $$:  
  $$
  x = 1 	imes 2 	imes 3 	imes 4 = 24, \quad n = 1 + x = 25
  $$  
  Here, $$ n = 25 $$, which is not prime.

- For the integers $$ 2, 3, 4, 5 $$:  
  $$
  x = 2 	imes 3 	imes 4 	imes 5 = 120, \quad n = 1 + x = 121
  $$  
  Here, $$ n = 121 $$, which is not prime.

Since there are cases where $$ n $$ is not prime, **$$ n $$ is not always prime**.

---

### Step 3: Check if $$ n $$ is a Perfect Square  
Let the four consecutive integers be represented as $$ m, m+1, m+2, m+3 $$. Then:  
$$
x = m(m+1)(m+2)(m+3)
$$  
We can rewrite this as:  
$$
x + 1 = (m^2 + 3m)(m^2 + 3m + 2) + 1
$$  
This simplifies to:  
$$
n = (m^2 + 3m + 1)^2
$$  
Thus, **$$ n $$ is always a perfect square**.

---

### Conclusion:  
- Statement I: True ($$ n $$ is odd).  
- Statement II: False ($$ n $$ is not always prime).  
- Statement III: True ($$ n $$ is a perfect square).

The correct answer is **I and III only**.

Answer

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