What is the remainder when `2(8!)-21(6!)` divides `14(7!)+14(13!)`?

Options

A. 1
B. 7!
C. 8!
D. 9!

Answer

To solve the problem of finding the remainder when \(2(8!) - 21(6!)\) divides \(14(7!) + 14(13!)\), we will follow these steps:

### Step 1: Simplify the expressions

1. **Calculate \(2(8!)\)**:
\[ 2(8!) = 2 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 2 \times 40320 = 80640 \]

2. **Calculate \(21(6!)\)**:
\[ 21(6!) = 21 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 21 \times 720 = 15120 \]

3. **Combine the two results**:
\[ 2(8!) - 21(6!) = 80640 - 15120 = 65520 \]

### Step 2: Simplify the divisor

1. **Calculate \(14(7!)\)**:
\[ 14(7!) = 14 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 14 \times 5040 = 70560 \]

2. **Calculate \(14(13!)\)**:
\[ 14(13!) = 14 \times (13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7!) = 14 \times 6227020800 = 86956867200 \]

3. **Combine the two results**:
\[ 14(7!) + 14(13!) = 70560 + 86956867200 \]

### Step 3: Find the remainder

1. **Now we need to find the remainder of \(65520\) when divided by \(70560 + 86956867200\)**:
- Since \(70560 + 86956867200\) is much larger than \(65520\), we can directly see that \(65520\) is less than the divisor.
- Therefore, the remainder is simply \(65520\).

### Step 4: Final calculation

1. **Now we need to find the remainder when \(65520\) is divided by \(70560 + 86956867200\)**:
\[ \text{Remainder} = 65520 \]

### Conclusion
The final answer is that the remainder when \(2(8!) - 21(6!)\) divides \(14(7!) + 14(13!)\) is \(65520\).