Divide 8 by 6. 8 / 6 = 1 with a remainder of 2.

Here are the solutions to the division problems:

1. $8640 \div 6$

   Step 1: Divide 8 by 6. $8 \div 6 = 1$ with a remainder of 2.

   Step 2: Bring down 6 to make 26. Divide 26 by 6. $26 \div 6 = 4$ with a remainder of 2.

   Step 3: Bring down 4 to make 24. Divide 24 by 6. $24 \div 6 = 4$ with a remainder of 0.

   Step 4: Bring down 0 to make 0. Divide 0 by 6. $0 \div 6 = 0$.

   $$8640 \div 6 = \boxed{\text{1440}}$$

2. $9720 \div 9$

   Step 1: Divide 9 by 9. $9 \div 9 = 1$ with a remainder of 0.

   Step 2: Bring down 7 to make 7. Divide 7 by 9. $7 \div 9 = 0$ with a remainder of 7.

   Step 3: Bring down 2 to make 72. Divide 72 by 9. $72 \div 9 = 8$ with a remainder of 0.

   Step 4: Bring down 0 to make 0. Divide 0 by 9. $0 \div 9 = 0$.

   $$9720 \div 9 = \boxed{\text{1080}}$$

3. $7350 \div 5$

   Step 1: Divide 7 by 5. $7 \div 5 = 1$ with a remainder of 2.

   Step 2: Bring down 3 to make 23. Divide 23 by 5. $23 \div 5 = 4$ with a remainder of 3.

   Step 3: Bring down 5 to make 35. Divide 35 by 5. $35 \div 5 = 7$ with a remainder of 0.

   Step 4: Bring down 0 to make 0. Divide 0 by 5. $0 \div 5 = 0$.

   $$7350 \div 5 = \boxed{\text{1470}}$$

4. $8824 \div 8$

   Step 1: Divide 8 by 8. $8 \div 8 = 1$ with a remainder of 0.

   Step 2: Bring down 8 to make 8. Divide 8 by 8. $8 \div 8 = 1$ with a remainder of 0.

   Step 3: Bring down 2 to make 2. Divide 2 by 8. $2 \div 8 = 0$ with a remainder of 2.

   Step 4: Bring down 4 to make 24. Divide 24 by 8. $24 \div 8 = 3$ with a remainder of 0.

   $$8824 \div 8 = \boxed{\text{1103}}$$

5. $9180 \div 6$

   Step 1: Divide 9 by 6. $9 \div 6 = 1$ with a remainder of 3.

   Step 2: Bring down 1 to make 31. Divide 31 by 6. $31 \div 6 = 5$ with a remainder of 1.

   Step 3: Bring down 8 to make 18. Divide 18 by 6. $18 \div 6 = 3$ with a remainder of 0.

   Step 4: Bring down 0 to make 0. Divide 0 by 6. $0 \div 6 = 0$.

   $$9180 \div 6 = \boxed{\text{1530}}$$

6. $7569 \div 3$

   Step 1: Divide 7 by 3. $7 \div 3 = 2$ with a remainder of 1.

   Step 2: Bring down 5 to make 15. Divide 15 by 3. $15 \div 3 = 5$ with a remainder of 0.

   Step 3: Bring down 6 to make 6. Divide 6 by 3. $6 \div 3 = 2$ with a remainder of 0.

   Step 4: Bring down 9 to make 9. Divide 9 by 3. $9 \div 3 = 3$ with a remainder of 0.

   $$7569 \div 3 = \boxed{\text{2523}}$$

7. $8400 \div 7$

   Step 1: Divide 8 by 7. $8 \div 7 = 1$ with a remainder of 1.

   Step 2: Bring down 4 to make 14. Divide 14 by 7. $14 \div 7 = 2$ with a remainder of 0.

   Step 3: Bring down 0 to make 0. Divide 0 by 7. $0 \div 7 = 0$.

   Step 4: Bring down 0 to make 0. Divide 0 by 7. $0 \div 7 = 0$.

   $$$$