1. Solve the following (2 marks each)

a) What is $84 \mod 9$?

To solve $84 \mod 9$, divide 84 by 9 and find the remainder. $84 \div 9 = 9 \text{ with remainder } 3$ Thus, $84 \mod 9 = 3$.

b) What is $52 \mod 5$?

To solve $52 \mod 5$, divide 52 by 5 and find the remainder. $52 \div 5 = 10 \text{ with remainder } 2$ Thus, $52 \mod 5 = 2$.

c) What is $-4 \mod 10$?

To solve $-4 \mod 10$, we need to adjust the negative number to find a positive remainder. The rule is that the remainder must be between 0 and 9. $-4 + 10 = 6$ Thus, $-4 \mod 10 = 6$.

2. Create a multiplication and addition table for modulo 7 (3 marks each)

For modulo 7, all calculations are done using remainders when divided by 7. Here's the addition and multiplication table:

Addition Table (mod 7):

+ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 0 \\ 2 & 2 & 3 & 4 & 5 & 6 & 0 & 1 \\ 3 & 3 & 4 & 5 & 6 & 0 & 1 & 2 \\ 4 & 4 & 5 & 6 & 0 & 1 & 2 & 3 \\ 5 & 5 & 6 & 0 & 1 & 2 & 3 & 4 \\ 6 & 6 & 0 & 1 & 2 & 3 & 4 & 5 \\ \end{array}$$ **Multiplication Table (mod 7):** $$\begin{array}{c|ccccccc} \times & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 0 & 2 & 4 & 6 & 1 & 3 & 5 \\ 3 & 0 & 3 & 6 & 2 & 5 & 1 & 4 \\ 4 & 0 & 4 & 1 & 5 & 2 & 6 & 3 \\ 5 & 0 & 5 & 3 & 1 & 6 & 4 & 2 \\ 6 & 0 & 6 & 5 & 4 & 3 & 2 & 1 \\ \end{array}$$ ### 3. It is 8:00 AM in a 24-hour world. What time is it in a 3-hour world? (2 marks) To solve this, we compute $$8 \mod 3$$. Divide 8 by 3: $$8 \div 3 = 2 \text{ with remainder } 2$$ Thus, in a 3-hour world, it would be 2:00 AM. ### 4. What residue classes exist for $$m = 4$$? (2 marks) For $$m = 4$$, the residue classes are the set of integers modulo 4. These are the possible remainders when divided by 4: $$\{ 0, 1, 2, 3 \}$$ Thus, the residue classes are $$[0], [1], [2], [3]$$. ### 5. Let $$m = 5$$, how many residue classes exist? State five integers in each class. (7 marks) For $$m = 5$$, there are 5 residue classes, corresponding to the remainders 0 through 4: $$\{ 0, 1, 2, 3, 4 \}$$ **Residue classes:** - Class $$[0]$$: Integers such that $$a \equiv 0 \mod 5$$. Some examples: $$0, 5, 10, 15, 20$$. - Class $$[1]$$: Integers such that $$a \equiv 1 \mod 5$$. Some examples: $$1, 6, 11, 16, 21$$. - Class $$[2]$$: Integers such that $$a \equiv 2 \mod 5$$. Some examples: $$2, 7, 12, 17, 22$$. - Class $$[3]$$: Integers such that $$a \equiv 3 \mod 5$$. Some examples: $$3, 8, 13, 18, 23$$. - Class $$[4]$$: Integers such that $$a \equiv 4 \mod 5$$. Some examples: $$4, 9, 14, 19, 24$$. --- ### Any questions on the solutions or concepts? Here are five related questions: 1. How do you solve $$75 \mod 6$$? 2. Can you explain the relationship between modular arithmetic and clock arithmetic? 3. How would you calculate $$-7 \mod 12$$? 4. What is the significance of residue classes in number theory? 5. How would you create a multiplication table for modulo 6? **Tip:** When working with negative numbers in modular arithmetic, always add the modulus to the number until you get a positive remainder.