$(x^n − y^n) ext{ is divisible by } (x−y) ext{, then the value of n is :}$

Q: $(x^n − y^n) \text{ is divisible by } (x−y) \text{, then the value of n is :}$

$(x^n − y^n) \text{ is divisible by } (x−y) \text{, then the value of n is :}$

Question

$(x^n − y^n) 	ext{ is divisible by } (x−y) 	ext{, then the value of n is :}$

Accepted Answer

The problem states that $ (x^n - y^n) $ is divisible by $ (x-y) $. We need to determine the value of $ n $ that satisfies this condition, with the correct answer being Option 4: a natural number.

### Explanation Supporting Option 4:

1. **Understanding the Divisibility:**
   - The expression $ (x^n - y^n) $ being divisible by $ (x-y) $ is a well-known result in algebra. This is based on the factorization of the difference of powers.
   - According to the polynomial division theorem, $ (x^n - y^n) $ can be expressed as $ (x-y)Q(x, y) $ where $ Q(x, y) $ is a polynomial in $ x $ and $ y $. This holds true for any natural number $ n $.

2. **Natural Numbers:**
   - Natural numbers are the set of positive integers starting from 1, i.e., $ \{1, 2, 3, \ldots\} $.
   - For any natural number $ n $, the expression $ (x^n - y^n) $ can be factored as $ (x-y)(x^{n-1} + x^{n-2}y + \cdots + y^{n-1}) $, which confirms the divisibility by $ (x-y) $.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: An even number**
   - While even numbers are a subset of natural numbers, the condition $ (x^n - y^n) $ being divisible by $ (x-y) $ is not restricted to even numbers. It holds for all natural numbers, not just even ones.

2. **Option 2: An odd number**
   - Similar to even numbers, odd numbers are also a subset of natural numbers. The divisibility condition is not exclusive to odd numbers; it applies to all natural numbers.

3. **Option 3: A prime number**
   - Prime numbers are specific natural numbers greater than 1 that have no divisors other than 1 and themselves. The divisibility condition does not require $ n $ to be prime. It holds for any natural number, regardless of whether it is prime or not.

### Conclusion:

The correct answer is Option 4: a natural number, because the condition $ (x^n - y^n) $ being divisible by $ (x-y) $ is satisfied for any natural number $ n $. This includes all positive integers, not limited to even, odd, or prime numbers specifically.