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Help me !!! Please

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Solution

Here are the answers to the addition table of Z5

0, 1, 2, 3, 4
1, 2, 3, 4, 0
2, 3, 4, 0, 1
3, 4, 0, 1, 2
4, 0, 1, 2, 3

Here is the anwer to the multiplication table of Z5:

0, 0, 0, 0, 0
0, 1, 2, 3, 4
0, 2, 4, 1, 3
0, 3, 1, 4, 2
0, 4, 3, 2, 1

Here is the answer to the addition table of Z8
0, 1, 2, 3, 4, 5, 6, 7
1, 2, 3, 4, 5, 6, 7, 0
2, 3, 4, 5, 6, 7, 0, 1
3, 4, 5, 6, 7, 0, 1, 2
4, 5, 6, 7, 0, 1, 2, 3
5, 6, 7, 0, 1, 2, 3, 4
6, 7, 0, 1, 2, 3, 4, 5,
7, 0, 1, 2, 3, 4, 5, 6

Here is the multiplication table for Z8:
0, 0, 0, 0, 0, 0, 0, 0
0, 1, 2, 3, 4, 5, 6, 7
0, 2, 4, 6, 1, 3, 4, 6
0, 3, 6, 1, 4, 1, 4, 0
0, 4, 0, 4, 0, 4, 0, 4
0, 5, 2, 7, 4, 1, 6, 3
0, 6, 4, 2, 0, 6, 4, 2
0, 7, 6, 5, 4, 3, 2, 1

When it comes to subtraction bein a well defiend operation is Z5 or Z8, it is not possible since x= y+z; As long as X>= Y this notion will hold but when X<Y then Z will come out ot be negative and negative values of Z cannot be a part of Z5 or Z8; Consider the example of (x,y)= (2,3) then 2= 3+ z so z= -1 but -1 is not a part of Z5 or Z8

About division too, the quotient will be a part of Z5 or Z8 only when the remainder is zero, Say x= 5 and Y=3 the quotient will be 1 and remainder will be 2 or the exact quotient will be 1.67 and 1.67 is not a part of Z5 or Z8 set;