The document discusses modular arithmetic and provides examples of addition, subtraction, and finding remainders when dividing numbers by various moduli (9, 4, 6). It shows that when dividing numbers by the modulus, the remainders repeat in a cyclic pattern from 0 back to the modulus - 1. For example, when dividing by 9, the remainders are 0, 1, 2, 3, 4, 5, 6, 7, 8 before repeating at 0 again.

1. Modular Arithmetic
Addition and Subtraction

2. Modulus 9
0 mod 9 = 0
1 mod 9 = 1
2 mod 9 = 2
3 mod 9 = 3
4 mod 9 = 4
5 mod 9 = 5
6 mod 9 = 6
7 mod 9 = 7
8 mod 9 = 8
0
1
2
3
5
6
8
7
4
9 mod 9 = 0
10 mod 9 = 1
11 mod 9 = 2
12 mod 9 = 3
13 mod 9 = 4
14 mod 9 = 5
15 mod 9 = 6
16 mod 9 = 7
17 mod 9 = 8

3. Modulus 9
18 mod 9 = 0
19 mod 9 = 1
20 mod 9 = 2
21 mod 9 = 3
22 mod 9 = 4
23 mod 9 = 5
24 mod 9 = 6
25 mod 9 = 7
26 mod 9 = 8
27 mod 9 = 0
28 mod 9 = 1
29 mod 9 = 2
30 mod 9 = 3
31 mod 9 = 4
32 mod 9 = 5
33 mod 9 = 6
34 mod 9 = 7
35 mod 9 = 8
0
1
2
3
5
6
8
7
4

4. Modulus 9
-9 mod 9 = 0
-8 mod 9 = 1
-7 mod 9 = 2
-6 mod 9 = 3
-5 mod 9 = 4
-4 mod 9 = 5
-3 mod 9 = 6
-2 mod 9 = 7
-1 mod 9 = 8
0
1
2
3
5
6
8
7
4

5. Modulus 9
(0 + 1) mod 9 = 1
(1 + 5) mod 9 = 6
(8 + 3) mod 9 = 11 mod 9 = 2
(15 + 13) mod 9 = 28 mod 9 = 1
= 15 mod 9 + 13 mod 9
= (6 + 4) mod 9 = 10 mod 9 = 1
0
1
2
3
5
6
8
7
4

6. Modulus 9
(7 - 2) mod 9 = 5
(16 - 5) mod 9 = 11 mod 9 = 2
= 16 mod 9 - 5 mod 9
= (7 - 5) mod 9 = 2
0
1
2
3
5
6
8
7
4

7. Modulus 4
0 mod 4 = 0
1 mod 4 = 1
2 mod 4 = 2
3 mod 4 = 3
4 mod 4 = 0
5 mod 4 = 1
12 mod 4 = 0
13 mod 4 = 1
0
1
2
3

8. Modulus 6
0 mod 6 = 0
1 mod 6 = 1
2 mod 6 = 2
3 mod 6 = 3
4 mod 6 = 4
5 mod 6 = 5
6 mod 6 = 0
7 mod 6 = 1
8 mod 6 = 2
0
1
2
3
5
4