`(103)^86-(86)^103` is divisible by

Options

A. 7
B. 13
C. 17
D. 23

Answer

To determine whether \( (103)^{86} - (86)^{103} \) is divisible by a certain number, we can use the Binomial Theorem. Let's break down the solution step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ (103)^{86} - (86)^{103} \]
We can express \( 103 \) and \( 86 \) in a form suitable for applying the Binomial Theorem:
\[ (103)^{86} = (1 + 102)^{86} \]
\[ (86)^{103} = (1 + 85)^{103} \]

### Step 2: Apply the Binomial Theorem
Using the Binomial Theorem, we can expand both expressions:
\[ (1 + 102)^{86} = \sum_{k=0}^{86} \binom{86}{k} (102)^k \]
\[ (1 + 85)^{103} = \sum_{j=0}^{103} \binom{103}{j} (85)^j \]

### Step 3: Identify the first few terms
We can focus on the first few terms of each expansion:
- For \( (1 + 102)^{86} \):
- The first term is \( 1 \)
- The second term is \( \binom{86}{1} \cdot 102 = 86 \cdot 102 \)

- For \( (1 + 85)^{103} \):
- The first term is \( 1 \)
- The second term is \( \binom{103}{1} \cdot 85 = 103 \cdot 85 \)

### Step 4: Subtract the expansions
Now, we subtract the two expansions:
\[ (1 + 102)^{86} - (1 + 85)^{103} = \left(1 + 86 \cdot 102 + \cdots\right) - \left(1 + 103 \cdot 85 + \cdots\right) \]
The \( 1 \) terms cancel out:
\[ 86 \cdot 102 - 103 \cdot 85 + \text{(higher order terms)} \]

### Step 5: Simplify the expression
Now, we simplify the first two terms:
\[ 86 \cdot 102 - 103 \cdot 85 = 8772 - 8755 = 17 \]
Thus, we have:
\[ (103)^{86} - (86)^{103} = 17 + \text{(higher order terms)} \]

### Step 6: Check divisibility
Since \( 17 \) is a constant and the higher order terms will also be multiples of \( 17 \) (as they involve combinations of \( 102 \) and \( 85 \), which are both divisible by \( 17 \)), we conclude that:
\[ (103)^{86} - (86)^{103} \text{ is divisible by } 17 \]

### Final Answer
Thus, \( (103)^{86} - (86)^{103} \) is divisible by \( 17 \).

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