Show that `7^(n) +5` is divisible by 6, where n is a positive integer

Answer

To show that \( 7^n + 5 \) is divisible by 6 for any positive integer \( n \), we can follow these steps:

### Step 1: Analyze \( 7^n \) modulo 6
First, we need to find the value of \( 7^n \) modulo 6. We can simplify \( 7 \) as follows:
\[ 7 \equiv 1 \mod 6 \]
This means that \( 7^n \) can be expressed as:
\[ 7^n \equiv 1^n \equiv 1 \mod 6 \]

### Step 2: Add 5 to \( 7^n \)
Now, we add 5 to \( 7^n \):
\[ 7^n + 5 \equiv 1 + 5 \mod 6 \]

### Step 3: Simplify the expression
Next, we simplify the expression:
\[ 1 + 5 = 6 \]
Thus, we have:
\[ 7^n + 5 \equiv 6 \mod 6 \]

### Step 4: Conclusion
Since \( 6 \equiv 0 \mod 6 \), we conclude that:
\[ 7^n + 5 \text{ is divisible by } 6 \]

### Final Result
Therefore, we have shown that \( 7^n + 5 \) is divisible by 6 for any positive integer \( n \).

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