If `n in 1,3,5,7,…` etc., then the value of `19^(n) - 23^n - 43^n + 47^n` is necessarily divisible by :

Options

A. 264
B. 246
C. 76
D. 129

Answer

To solve the problem, we need to evaluate the expression \( 19^n - 23^n - 43^n + 47^n \) for odd values of \( n \) (i.e., \( n = 1, 3, 5, 7, \ldots \)) and determine what it is necessarily divisible by.

### Step 1: Substitute a value for \( n \)
Let's start by substituting \( n = 3 \) into the expression:

\[ 19^3 - 23^3 - 43^3 + 47^3 \]

### Step 2: Calculate each term
Now we calculate each term separately:

1. \( 19^3 = 6859 \)
2. \( 23^3 = 12167 \)
3. \( 43^3 = 79507 \)
4. \( 47^3 = 103823 \)

### Step 3: Substitute the calculated values into the expression
Now substitute these values back into the expression:

\[ 6859 - 12167 - 79507 + 103823 \]

### Step 4: Simplify the expression
Now, we perform the calculations step by step:

1. Calculate \( 6859 - 12167 = -5308 \)
2. Then, calculate \( -5308 - 79507 = -84815 \)
3. Finally, calculate \( -84815 + 103823 = 19008 \)

So, the value of the expression when \( n = 3 \) is:

\[ 19^3 - 23^3 - 43^3 + 47^3 = 19008 \]

### Step 5: Check divisibility
Now we need to check if \( 19008 \) is divisible by the options given:

1. **Check divisibility by 264**:
\[ 19008 \div 264 = 72 \]
This is an integer, so \( 19008 \) is divisible by \( 264 \).

2. **Check divisibility by 246**:
\[ 19008 \div 246 \approx 77.3 \]
This is not an integer.

3. **Check divisibility by 76**:
\[ 19008 \div 76 \approx 250.1 \]
This is not an integer.

4. **Check divisibility by 129**:
\[ 19008 \div 129 \approx 147.3 \]
This is not an integer.

### Conclusion
The only option that \( 19008 \) is divisible by is \( 264 \). Therefore, the answer is:

\[ \text{The value of } 19^n - 23^n - 43^n + 47^n \text{ is necessarily divisible by } 264. \]