Let's solve the given problems based on the modulus operation explanation in your image.

We are asked to calculate the quotient and the remainder (modulus) for several division problems. Here's the breakdown:

Example Formula:

For any integers $a$ and $b$, we have: $a = bq + r \quad \text{where} \quad 0 \leq r < b$ This means the remainder $r$, when dividing $a$ by $b$, must be non-negative and less than $b$.

Solving the Given Problems:

1. The quotient of $25 \div 6$:

   • $25 = 6 \times 4 + 1$
   • Quotient = 4, $25 \mod 6 = 1$

2. The quotient of $27 \div 9$:

   • $27 = 9 \times 3 + 0$
   • Quotient = 3, $27 \mod 9 = 0$

3. The quotient of $22 \div 10$:

   • $22 = 10 \times 2 + 2$
   • Quotient = 2, $22 \mod 10 = 2$

4. The quotient of $21 \div 10$:

   • $21 = 10 \times 2 + 1$
   • Quotient = 2, $21 \mod 10 = 1$

5. The quotient of $22 \div 7$:

   • $22 = 7 \times 3 + 1$
   • Quotient = 3, $22 \mod 7 = 1$

6. The quotient of $27 \div 8$:

   • $27 = 8 \times 3 + 3$
   • Quotient = 3, $27 \mod 8 = 3$

Now, for the larger numbers:

7. $618432955 \mod 101$:

   • Quotient: $618432955 \div 101 \approx 6122148$
   • Remainder: $618432955 - (6122148 \times 101) = 27$
   • $618432955 \mod 101 = 27$

8. $345828498 \mod 101$:

   • Quotient: $345828498 \div 101 \approx 3424044$
   • Remainder: $345828498 - (3424044 \times 101) = 54$
   • $345828498 \mod 101 = 54$

Let me know if you'd like further details or explanations on any of these! Here are five follow-up questions:

1. How do you determine the remainder in modulus operations?
2. What is the significance of the non-negative remainder in the modulus operation?
3. Can you find the modulus of $100 \div 7$?
4. Why does $a \mod b$ always produce a result in the range $0 \leq r < b$?
5. How does the division algorithm help in finding the modulus?

Tip: Always remember to first find the quotient, and then compute the remainder using $r = a - bq$.