Arithmetic properties of $\mathbb{Z}/n\mathbb{Z}$: a summary

In this section, we give an abstract summary of the main results discussed in this chapter.

Fix an integer modulus $n>1$. Recall that if $a$ is any integer then $\overline{a}=a+ n\mathbb{Z}$. Let

$$\mathbb{Z}/n\mathbb{Z}=\{ \overline{0},\overline{1},..., \overline{n-1}\},$$

denote the set of all residue classes of $n$. Define addition $+$ and multiplication $\cdot$ in $\mathbb{Z}/n\mathbb{Z}$ by the following rules:

$$(a+ n\mathbb{Z})+(b+ n\mathbb{Z})=a+b+ n\mathbb{Z},$$

and

$$(a+ n\mathbb{Z})\cdot (b+ n\mathbb{Z})=a\cdot b+ n\mathbb{Z}.$$

The properties of addition and multiplication are summarized in the following proposition.

Proposition 1.9.1   Fix an integer modulus $n>1$. For any integers $a,b,c$,

1. $\overline{a}+\overline{b}=\overline{a+b} =\overline{b}+\overline{a}$, (``addition is commutative'')
2. $\overline{a}\overline{b}=\overline{ab} =\overline{b}\overline{a}$, (``multiplication is commutative'')
3. $(\overline{a}+\overline{b})+\overline{c} =\overline{a}+(\overline{b}+\overline{c})$, (``addition is associative'')
4. $(\overline{a}\overline{b})\overline{c} =\overline{a}(\overline{b}\overline{c})$, (``multiplication is associative'')
5. $(\overline{a}+\overline{b})\overline{c}= \overline{a}\overline{c}+ \overline{b}\overline{c}$, (``distributive'')
6. $\overline{a}\overline{1}=\overline{a} =\overline{1}\overline{a}$, (`` $\overline{1}$ is a multiplicative identity'')
7. $\overline{a}+\overline{-a}=\overline{0}$,
8. $\overline{a}\overline{0}=\overline{0}$,
9. $\overline{a}+\overline{0}=\overline{a}$ (`` $\overline{0}$ is a additive identity''),
10. if $\overline{a}\overline{c}=\overline{b}\overline{c}$ and $gcd(a,c)=1$ then $\overline{a}=\overline{b}$ (``cancellation law''),
11. if $gcd(a,n)=1$ then $\overline{a}^{\phi(n)}=\overline{1}$ (``Euler's theorem''),
12. $gcd(a,n)=1$ if and only if there is an element, denoted $\overline{a}^{-1}$, such that $\overline{a}\cdot \overline{a}^{-1}= \overline{1}$,
13. there is a solution $\overline{x}$ to $\overline{a}\cdot \overline{x}=\overline{b}$ if and only if $gcd(a,n)\vert b$.

All these are restatements, in different notation, of results proven above. The verification of all these are left as an exercise.

In the special case when $n$ is a prime $p$, we have the following facts.

Proposition 1.9.2   Fix a prime modulus $p>1$. For any integers $a,b,c$,

1. $\overline{a}+\overline{b}=\overline{a+b} =\overline{b}+\overline{a}$, (``addition is commutative'')
2. $\overline{a}\overline{b}=\overline{ab} =\overline{b}\overline{a}$, (``multiplication is commutative'')
3. $(\overline{a}+\overline{b})+\overline{c} =\overline{a}+(\overline{b}+\overline{c})$, (``addition is associative'')
4. $(\overline{a}\overline{b})\overline{c} =\overline{a}(\overline{b}\overline{c})$, (``multiplication is associative'')
5. $(\overline{a}+\overline{b})\overline{c}= \overline{a}\overline{c}+ \overline{b}\overline{c}$, (``distributive'')
6. $\overline{a}\overline{1}=\overline{a} =\overline{1}\overline{a}$, (`` $\overline{1}$ is a multiplicative identity'')
7. $\overline{a}+\overline{-a}=\overline{0}$,
8. $\overline{a}\overline{0}=\overline{0}$,
9. $\overline{a}+\overline{0}=\overline{a}$ (`` $\overline{0}$ is a additive identity''),
10. if $\overline{a}\overline{c}=\overline{b}\overline{c}$ and $\overline{c}\not= \overline{0}$ then $\overline{a}=\overline{b}$ (``cancellation law''),
11. $\overline{a}^{p}=\overline{a}$ (``Fermat's little theorem''),
12. if $\overline{a}\not= \overline{0}$ then there is an element, denoted $\overline{a}^{-1}$, such that $\overline{a}\cdot \overline{a}^{-1}= \overline{1}$.