
3.2: Residue Systems and Euler’s φ-Function

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residue of $$a$$ modulo $$m$$. As a result, we see that any integer is congruent to one of the integers $$0,1,2,...,m-1$$ modulo m.

A complete residue system modulo $$m$$ is a set of integers such that every integer is congruent modulo $$m$$ to exactly one integer of the set.

The easiest complete residue system modulo $$m$$ is the set of integers $$0,1,2,...,m-1$$. Every integer is congruent to one of these integers modulo $$m$$.

The set of integers $$\{0,1,2,3,4\}$$ form a complete residue system modulo $$5$$. Another complete residue system modulo $$5$$ could be $$6,7,8,9,10$$.

A reduced residue system modulo $$m$$ is a set of integers $$r_i$$ such that $$(r_i,m)=1$$ for all $$i$$ and $$r_i \neq r_j (mod \ m)$$ if $$i\neq j$$.

Notice that, a reduced residue system modulo $$m$$ can be obtained by deleting all the elements of the complete residue system set that are not relatively prime to $$m$$.

The set of integers $$\{1,5\}$$ is a reduced residue system modulo $$6$$.

The following lemma will help determine a complete residue system modulo any positive integer $$m$$.

LEmma

A set of $$m$$ incongruent integers modulo $$m$$ forms a complete residue system modulo $$m$$.

We will prove this lemma by contradiction. Suppose that the set of $$m$$ integers does not form a complete residue system modulo $$m$$. Then we can find at least one integer $$a$$ that is not congruent to any element in this set. Hence non of the elements of this set is actually congruent to the remainder when $$a$$ is divided by $$m$$. Thus dividing by $$m$$ yields to at most $$m-1$$ remainders. Therefore by the pigeonhole principle, at least two integers in the set that have the same remainder modulo $$m$$. This is a contradiction since the set of integers is formed of $$m$$ integers that are incongruent modulo $$m$$.

If $$a_1, a_2,...,a_m$$ is a complete residue system modulo $$m$$, and if $$k$$ is a positive integer with $$(k,m)=1$$, then $ka_1+b, ka_2+b,...,ka_m+b$ is another complete residue system modulo $$m$$ for any integer $$b$$.

Let us prove first that no two elements of the set $$\{ka_1+b, ka_2+b,...,ka_m+b\}$$ are congruent modulo $$m$$. Suppose there exists $$i$$ and $$j$$ such that $ka_i+b\equiv ka_j+b(mod\ m).$ Thus we get that $ka_i\equiv ka_j(mod \ m).$ Now since $$(k,m)=1$$, we get $a_i\equiv a_j(mod\ m)$ But for $$i\neq j$$, $$a_i$$ is inequivalent to $$a_j$$ modulo $$m$$. Thus $$i=j$$. Now notice that there are $$m$$ inequivalent integers modulo m and thus by Lemma 10, the set form a complete residue system modulo $$m$$.

Euler’s $$\phi$$-Function

We now present a function that counts the number of positive integers less than a given integer that are relatively prime to that given integer. This function is called Euler $$\phi$$-function. We will discuss the properties of Euler $$\phi$$-function in details in chapter 5. It will be sufficient for our purposes in this chapter to the notation.

The Euler $$\phi$$-function of a positive integer n, denoted by $$\phi(n)$$ counts the number of positive integers less than $$n$$ that are relatively prime to n.

Since 1 and 3 are the only two integers that are relatively prime to 4 and less than 4, then $$\phi(4)=2$$. Also, 1,2,...,6 are the integers that are relatively prime to 7 that are less than 7, thus $$\phi(7)=6$$.

Now we can say that the number of elements in a reduced residue system modulo $$n$$ is $$\phi(n)$$.

If $$a_1,a_2,...,a_{\phi(n)}$$ is a reduced residue system modulo $$n$$ and $$(k,n)=1$$, then $$ka_1,ka_2,...,ka_{\phi(n)}$$ is a reduced residue system modulo $$n$$.

The proof proceeds exactly in the same way as that of Theorem 24.

Exercises

1. Give a reduced residue system modulo 12.

2. Give a complete residue system modulo 13 consisting only of odd integers.

3. Find $$\phi(8)$$ and $$\phi(101)$$.

Contributors

• Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.