The remainder when m + n is divided by 12 is 8, and the remainder when `m - n` is divided by 12 is 6. If `m > n`, then what is the remainder when mn divided by 6?

Answer

To solve the problem step by step, we start with the given conditions:

1. The remainder when \( m + n \) is divided by 12 is 8.
2. The remainder when \( m - n \) is divided by 12 is 6.
3. We know that \( m > n \).

### Step 1: Set up the equations based on the remainders.

From the first condition:
\[ m + n \equiv 8 \ (\text{mod} \ 12) \]
This can be expressed as:
\[ m + n = 12k + 8 \quad \text{for some integer } k \]

From the second condition:
\[ m - n \equiv 6 \ (\text{mod} \ 12) \]
This can be expressed as:
\[ m - n = 12j + 6 \quad \text{for some integer } j \]

### Step 2: Add the two equations.

Adding the two equations:
\[ (m + n) + (m - n) = (12k + 8) + (12j + 6) \]
This simplifies to:
\[ 2m = 12k + 12j + 14 \]
\[ 2m = 12(k + j) + 14 \]

### Step 3: Solve for \( m \).

Dividing both sides by 2 gives:
\[ m = 6(k + j) + 7 \]

### Step 4: Subtract the second equation from the first.

Now, subtract the second equation from the first:
\[ (m + n) - (m - n) = (12k + 8) - (12j + 6) \]
This simplifies to:
\[ 2n = 12k - 12j + 2 \]
\[ 2n = 12(k - j) + 2 \]

### Step 5: Solve for \( n \).

Dividing both sides by 2 gives:
\[ n = 6(k - j) + 1 \]

### Step 6: Find \( m \) and \( n \).

From the equations derived:
\[ m = 6(k + j) + 7 \]
\[ n = 6(k - j) + 1 \]

### Step 7: Calculate \( mn \).

Now, we need to find \( mn \):
\[ mn = (6(k + j) + 7)(6(k - j) + 1) \]

### Step 8: Find the remainder when \( mn \) is divided by 6.

To find the remainder when \( mn \) is divided by 6, we can simplify:
- The term \( 6(k + j) \) is divisible by 6.
- The term \( 6(k - j) \) is also divisible by 6.

Thus:
\[ mn \equiv (7 \cdot 1) \ (\text{mod} \ 6) \]
\[ mn \equiv 7 \ (\text{mod} \ 6) \]
Calculating \( 7 \mod 6 \):
\[ 7 \div 6 = 1 \quad \text{remainder } 1 \]

### Final Answer:
The remainder when \( mn \) is divided by 6 is **1**.