1.      Solve the following (2 marks each)

a)      What is 84 (mod 9)?
 
 
b)      What is 52 (mod 5)?
 
 
c)      What is -4 (mod 10)?
 
 
2.      Create a multiplication and addition table for modulo 7. (3 marks each)
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3.      It is 8:00AM in a 24 hour world. What time is it in a 3 hour world? (2 marks)
 
4.      What residue classes exists for m = 4? (2 marks)
 
5.      Let m = 5, how many residue classes exists. State five integers (*a*) in each of the classes. (7 marks)

### 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):**
\[
\begin{array}{c|ccccccc}
+ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
1 & 1 & 2 & 3 & 4 & 5 & 6 & 0 \\
2 & 2 & 3 & 4 & 5 & 6 & 0 & 1 \\
3 & 3 & 4 & 5 & 6 & 0 & 1 & 2 \\
4 & 4 & 5 & 6 & 0 & 1 & 2 & 3 \\
5 & 5 & 6 & 0 & 1 & 2 & 3 & 4 \\
6 & 6 & 0 & 1 & 2 & 3 & 4 & 5 \\
\end{array}
\]

**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: \( 4, 9, 14, 19, 24 \).

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### 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.

1. Solve the following (2 marks each)
a) What is 84 (mod 9)?
b) What is 52 (mod 5)?
c) What is -4 (mod 10)?
2. Create a multiplication and addition table for modulo 7. (3 marks each)
3. It is 8:00 AM in a 24-hour world. What time is it in a 3-hour world? (2 marks)
4. What residue classes exist for m = 4? (2 marks)
5. Let m = 5, how many residue classes exist. State five integers in each of the classes. (7 marks)

Solve modular arithmetic problems including modulus operations (e.g., 84 mod 9, 52 mod 5), create addition and multiplication tables for modulo 7, and explore residue classes for m = 4 and m = 5. Ideal for students in Grades 8-10.

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