60. Construct the addition table and the multiplication table for 7-arithmetic. List all factorizations of the numbers 0 and 1 into two factors.
60. Addition Table

|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 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 |

Multiplication Table

|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 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 |
|  |  |  |  |  |  |  |  |

Decompositions of zero and one are as follows:

\[
\begin{gathered}
0=0 \cdot 0=1 \cdot 0=2 \cdot 0=3 \cdot 0=4 \cdot 0=5 \cdot 0=6 \cdot 0 \\
1=1 \cdot 1=2 \cdot 4=3 \cdot 5=6 \cdot 6
\end{gathered}
\]

We notice that in 7-arithmetic, the product of two numbers can be zero only if at least one of the factors is zero. In this respect, 7-arithmetic is similar to ordinary arithmetic and differs from 10-arithmetic.

Let's compare the 7-arithmetic multiplication table with the multiplication table in 10-arithmetic (p. 51). There is a significant difference between these tables: in each row of the 7-arithmetic table (except the row corresponding to zero), all numbers from 7-arithmetic appear exactly once; in contrast, in the 10-arithmetic table, some numbers from 10-arithmetic are missing in certain rows, while others appear multiple times. Such rows with repetitions, besides the row for zero, are also the rows for \(2, 4, 5, 6, 8\), i.e., the rows corresponding to divisors of zero (see the solution to problem 57). The absence of rows with repetitions in the 7-arithmetic table is related to the absence of non-zero divisors of zero in 7-arithmetic.