For any natural number `n`, the number `A` given by `A=2903^(n)-803^(n)-464^(n)+261^(n)` is divisible by

Options

A. 7
B. 271
C. 1897
D. 13279

Answer

To determine the divisibility of the expression \( A = 2903^n - 803^n - 464^n + 261^n \) for any natural number \( n \), we can follow these steps:

### Step 1: Factor the expression
We can rearrange \( A \) as follows:
\[ A = (2903^n - 803^n) + (261^n - 464^n) \]

### Step 2: Apply the divisibility rule for differences of powers
Using the property that \( a^n - b^n \) is divisible by \( a - b \), we can analyze each part:
1. For \( 2903^n - 803^n \):
\[ 2903 - 803 = 2100 \]
Therefore, \( 2903^n - 803^n \) is divisible by \( 2100 \).

2. For \( 464^n - 261^n \):
\[ 464 - 261 = 203 \]
Therefore, \( 464^n - 261^n \) is divisible by \( 203 \).

### Step 3: Check divisibility of 2100 and 203 by 7
Now we need to check if \( 2100 \) and \( 203 \) are divisible by \( 7 \):
- \( 2100 \div 7 = 300 \) (which is an integer, so \( 2100 \) is divisible by \( 7 \)).
- \( 203 \div 7 = 29 \) (which is also an integer, so \( 203 \) is divisible by \( 7 \)).

Thus, both parts of \( A \) are divisible by \( 7 \), leading to:
\[ A \text{ is divisible by } 7. \]

### Step 4: Check divisibility of 2903 and 464 by 271
Next, we check the divisibility of \( A \) by \( 271 \):
1. For \( 2903^n - 464^n \):
\[ 2903 - 464 = 2439 \]
We need to check if \( 2439 \) is divisible by \( 271 \):
\[ 2439 \div 271 = 9 \text{ (which is an integer, so } 2439 \text{ is divisible by } 271). \]

2. For \( 803^n - 261^n \):
\[ 803 - 261 = 542 \]
We need to check if \( 542 \) is divisible by \( 271 \):
\[ 542 \div 271 = 2 \text{ (which is an integer, so } 542 \text{ is divisible by } 271). \]

Thus, both parts of \( A \) are divisible by \( 271 \), leading to:
\[ A \text{ is divisible by } 271. \]

### Step 5: Conclusion on divisibility
Since \( A \) is divisible by both \( 7 \) and \( 271 \), it follows that:
\[ A \text{ is divisible by } 7 \times 271 = 1897. \]

### Final Answer
Thus, \( A \) is divisible by \( 1897 \).

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