a. 

The rules are similar. Multiplication is closed, commutative, and associative. There is always exactly one answer to a multiplication problem. Multiplying by 0 results in 0, and multiplying by 1 results in the original number. One difference is that sometimes, in a finite system, you can multiply the same (non-zero!) number by two different numbers and get the same answer (for example, 4 • 2 and 4 • 7 both equal 8).

b. 

The rules are not as similar. In the finite system, some division problems have no solution (74, for example), though this is not very different from whole numbers. However, some division problems have more than one solution (84, for example, which in units digit arithmetic can be solved by both 2 and 7). Therefore, you cannot divide by 0, 2, 4, 5, 6, or 8 in this system. In the real number system, we can divide by all non-zero numbers. Also, in the finite system, as in the real number system, you can't make sense of 0 divided by 0 because there are too many solutions, and you can't make sense of number a divided by 0 for any other number a, because there are no solutions.

a. 

Yes, it acts in the same manner.

b. 

The distributive law ties together addition and multiplication: a • (b + c) = (a • b) + (a • c). To consider a system with two operations, we must have a property that tells us how the two operations are related. Otherwise, we would not know how to compute an expression that contains both addition and multiplication.

Similarly, in the real number system, the distributive law allows us to see how these operations relate to each other. It also allows us to use different methods to compute products. For example, if you need to multiply 25 by 99, the distributive law allows you to do the computation as 25 • (100 - 1), which you can do in your head.