1. The problem is to construct an addition table modulo 9. 2. Modulo addition means adding two numbers and then taking the remainder when divided by 9. 3. The formula is $$a + b \equiv (a + b) \mod 9$$ where $a$ and $b$ are integers from 0 to 8. 4. We create a 9x9 table where rows and columns represent numbers 0 through 8. 5. Each cell at row $a$ and column $b$ contains the value $$(a + b) \mod 9$$. 6. For example, $7 + 5 = 12$, and $12 \mod 9 = 3$, so the cell at row 7, column 5 is 3. 7. The table is symmetric because addition is commutative. Here is the addition table modulo 9: | + | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---|---| | 0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 0 | | 2 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 0 | 1 | | 3 | 3 | 4 | 5 | 6 | 7 | 8 | 0 | 1 | 2 | | 4 | 4 | 5 | 6 | 7 | 8 | 0 | 1 | 2 | 3 | | 5 | 5 | 6 | 7 | 8 | 0 | 1 | 2 | 3 | 4 | | 6 | 6 | 7 | 8 | 0 | 1 | 2 | 3 | 4 | 5 | | 7 | 7 | 8 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | | 8 | 8 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |