Modulo Arithmetic Z_n.

Z_n is the set of remainders after division by n.

For example, Z₃ = {0, 1, 2}, Z₄ = {0, 1, 2, 3}, Z₅ = {0, 1, 2, 3, 4}etc.

Elements of e.g. Z₄ = {0, 1, 2, 3} can be added or multiplied. The results of, say, addition, are best set out in what is called a Cayley table.

Note that 3 + 3 = 2 since 2 is the remainder after division by 4.

Z₄ = {0, 1, 2, 3} forms a group under addition.

• Associativity is given.
• The table demonstrates closure.
• The identity element is 0.
• Every element has an inverse:

However, Z₄ = {0, 1, 2, 3}does not form a group under multiplication.
Associativity and closure work out fine. The identity is 1.
But 0 does not have an inverse. (Try it! Zero times what gives 1?.)
This failure prevents Z₄ from being a group under multiplication.

EXAMPLE

Assuming that the multiplication of integers, mod 8, is associative, show that the set
A = {1, 3, 5, 7}, mod 8
is a group under multiplication. (LCHM 1986 I 8a )

SOLUTION

• Associativity is given.
• The Cayley table shows closure.
• The identity element is 1.
• Every element has an inverse:

Thus A = {1, 3, 5, 7}, mod 8 is a group under multiplication.