(i)

Multiplication ⊗ table in modulo 9 on {2, 3, 7, 8}

⊗	2	3	7	8
2	4	6	5	7
3	6	0	3	6
7	5	3	4	2
8	7	6	2	1

(ii)

Truth set of 8 ⊗ m = 2

{m : m = 7}

Note: the question stated truth set hence just stating m = 7 is incorrect.

(i)

Row 2

1 + 1 = 2 and 2 mod 5 = 2

1 + 2 = 3 and 3 mod 5 = 3

1 + 3 = 4 and 4 mod 5 = 4

1 + 4 = 5 and 5 mod 5 = 0

Row 3

2 + 1 = 3 and 3 mod 5 = 3

2 + 2 = 4 and 4 mod 5 = 4

2 + 3 = 5 and 5 mod 5 = 0

2 + 4 = 6 and 6 mod 5 = 1

Row 4

3 + 1 = 4 and 4 mod 5 = 4

3 + 2 = 5 and 5 mod 5 = 0

3 + 3 = 6 and 6 mod 5 = 1

3 + 4 = 7 and 7 mod 5 = 2

Row 5

4 + 1 = 5 and 5 mod 5 = 0

4 + 2 = 6 and 6 mod 5 = 1

4 + 3 = 7 and 7 mod 5 = 2

4 + 4 = 8 and 8 mod 5 = 3

⊕	1	2	3	4
1	2	3	4	0
2	3	4	0	1
3	4	0	1	2
4	0	1	2	3

Row 2

1 x 1 = 1 and 1 mod 5 = 1

1 x 2 = 2 and 2 mod 5 = 2

1 x 3 = 3 and 3 mod 5 = 3

1 x 4 = 4 and 4 mod 5 = 4

Row 3

2 x 1 = 2 and 2 mod 5 = 2

2 x 2 = 4 and 4 mod 5 = 4

2 x 3 = 6 and 6 mod 5 = 1

2 x 4 = 8 and 8 mod 5 = 3

Row 4

3 x 1 = 3 and 3 mod 5 = 3

3 x 2 = 6 and 6 mod 5 = 1

3 x 3 = 9 and 9 mod 5 = 4

3 x 4 = 12 and 12 mod 5 = 2

Row 5

4 x 1 = 4 and 4 mod 5 = 4

4 x 2 = 8 and 8 mod 5 = 3

4 x 3 = 12 and 12 mod 5 = 2

4 x 4 = 16 and 16 mod 5 = 1

⊗	1	2	3	4
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1

(ii)