Other Mod Systems
1. Make an addition table for modular arithmetic mod 6.
+
0
1
2
3
4
5
0
1
2
3
4
5
2. Make a multiplication table for modular arithmetic mod 6.
x
0
1
2
3
4
5
0
1
2
3
4
5
3. List some patterns you see in (Z6, +, x). Compare and contrast (Z6, +, x) with
(Z7, +, x).
4. For each of the following statements, determine if it is True or False. If you
actually prove that a statement is true by considering all possibilities, write
proven next to it.
a. a + a = a for all a in Z6.
b. The set (Z6, +) has an identity element.
c. For each a in Z6, there is an element x in Z6 such that a + x = 0. [Note: if
a + x = 0 we will write x = −a and read “x is the additive inverse of a.”
d. There are two different elements x and y in Z6 such that 3 + x = 0 and 3 +
y = 0.
e. For every two elements a and b in Z6, a + b is an element of Z6.
f. −(−𝑎) = 𝑎 for all a in Z6 .
g. a + b = b + a for all elements a and b in Z6.
h. a + (−𝑏) = b + (−a) for all a, b in Z6.
i. − (a + b) = −a + −b for all a,b in Z6.
j. a + (b + c) = (a + b) + c for all elements a, b, c in Z6.
k. If a = −b then b = −a.
2 Other Mod Systems
5. For each of the following statements, decide whether it is True or False. If you
actually prove that a statement is true by considering all possibilities, write
proven.
There is an identity element in (Z6, ).
For any elements a, b in Z6, a  b is an element in Z6.
There is an x in Z6 such that x  0 = 1.
For all a, b, c in Z6, if a  b = a  c and a 0, then b = c.
a  b = a2  b2 for some a, b in Z6 such that a b.
a  b = b  a for all elements a and b in Z6.
a  (b  c) = (a  b)  c for all elements a, b, and c in Z6.
For each a in Z6, there is an element x in Z6 such that a  x = 1. [Note: if
a  x = 1 we will write x = 𝑎−1 and read “x is the multiplicative inverse of
a.”
i. There are two different elements x and y in Z6 such that
5  x = 1 and 5  y = 1.
j. For all non-zero elements a, b in Z6, if a = 𝑏 −1 then b = 𝑎−1 .
k. For all a, b in Z6, if a 0 and b  0, then a  b  0.
l. a  (b  𝑎−1 ) = b for all a, b in Z6 with a 0.
a.
b.
c.
d.
e.
f.
g.
h.
6. Use your tables to solve the following equations for x.
a. 5x  4 (mod 6)
b. 2x  0 (mod 6)
c. 5 + x  1 (mod 6)
d. 2x + 3  5 (mod 6)
e. 2x  3 (mod 6)
f. Does every equation have a solution mod 6? Explain.
7. Construct addition and multiplication tables for mods 3, 4, 5, and 8. (Your group
will be assigned two). Compare and contrast your systems with (Z, +, x) and with
(Z6, +, x) and (Z7, +, x) arithmetic. Is there a conjecture you can make about the
mathematical structure of the different mod systems? If so, see if you can
explain what happens and why.
4 Other Mod Systems