MÔN HỌC NUMBER THEORY ĐỀ TÀI PROBLEMS ABOUT CONGRUENCE OF THE FIRST DEGREE

ĐẠI HỌC THỦ ĐÔ HÀ NỘI KHOA SƯ PHẠM

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MÔN HỌC: NUMBER THEORY ĐỀ TÀI: PROBLEMS ABOUT

CONGRUENCE OF THE FIRST DEGREE

HANOI METROPOLITAN UNIVERSITY FACULTY OF PEDAGOGY

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SUBJECT: NUMBER THEORY TOPIC: PROBLEMS ABOUT

CONGRUENCE OF THE FIRST DEGREE

Supervisor : Nguyen Thi Hong Student : Tran Quynh Ngan

ID Student : 222000470

HaNoi, January 2024

ACKNOWLEDGEMENTS

In the process of doing my research paper, I have received a lot of experience, guidance and encouragement from my teachers and friends

To begin with, I would like to express my deepest gratitude to my supervisor Ms Nguyen Thi Hong, the lecturer of foreign language faculty, HaNoi Metropolitan University, for her whole-hearted guidance and support Without her valuable recommendations and advice, I could not finish this thesis successfully

HaNoi, January 2024

Tran Quynh Ngan

4 Object and range of study 4

5 Structure of the study 5

CHAPTER 1: CONGRUENCES OF THE FIRST DEGREE 6

1.1.Basic concepts 6

1.2.Equations in congruence 10

1.3.Congruences in the first degree 12

CHAPTER 2: SOME APPLICATIONS OF CONGRUENCES OF THE FIRST DEGREE 19

2.1 Basic skills 19

2.2 Higher – Order Application 20

2.3 Further explorations 23

2.3.1 Some application and selected exercises 23

2.3.2 Going farther: More Analysis of Divisibility Tests 25

REFRENCES 26

Number theory is a very important part of Mathematics, we can say that it is a foundation of Mathematics Carl Freidrich Gauss, a very famous mathematician, said: "Mathematics is the queen of science and arithmetic is the queen of mathematics Thus, we can see the importance of this subject As with so many concepts we will see, congruence is simple, perhaps familiar to you, yet enormously useful and powerful

in the study of number theory If n is a positive integer, we say the integers a and b are congruent modulo n, and write 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), if they have the same remainder on division by n (By remainder, of course, we mean the unique number r defined by the Division Algorithm.) This notation, and much of the elementary

theory of congruence, is due to the famous German mathematician, Carl Friedrich Gauss—certainly the outstanding mathematician of his time, and perhaps the greatest mathematician of all time

INTRODUCTION

1 Rationale

A congruence is simply a declaration of divisibility One of history's greatest mathematicians, Carl Friedrich Gauss (1777–1855), developed the notion of congruences Gauss made numerous notable contributions to the theory of numbers, including the fundamental concepts Although number theory had been studied somewhat systematically earlier by Pierre de Fermat (1601–1665), Gauss was the first to develop the field as a branch of mathematics rather than just a distributed collection of interesting issues

Gauss established the idea of congruences in his book Disquisitions Arithmetical,

which was written at the age of 24 and quickly became recognized as a key instrument for the study of number theory The theorems of Fermat and Euler are particularly important, which offer effective methods for examining the multiplicative properties of congruence These two number theory founders approached their work in quite different ways

Fermat, a lawyer by trade, enjoyed mathematics as a pastime Through correspondence with other mathematicians, he shared his mathematical concepts while providing very little information about the justifications for his claims (One of his statements is referred to as Fermat's "final theorem," despite the fact that it has never been proven and is, therefore, not a theorem at all.) On the other hand, Leonard Euler (1707–1783) covered nearly all of the known fields of mathematics throughout his lifetime.

2 Aims of the study

The main goal of the thesis is to study about the congruence of the first degree, the Gauss mathematician and some applications in Mathematics

3 Research questions

As a basis for my investigation, the following research questions were formulated: - What is the congruence of the first degree?

- What are the applications of the congruence of the first degree?

4 Object and range of study

- Object: The word congruent refers to similar or twin An object is congruent to another object when they are exactly the same in shape and size Two objects are congruent when they are the mirror image of each other Many theorems are connected with equality and congruency We say two lines are congruent only if they have equal length, two angles are congruent if they have an equal measure, and two triangles are congruent if they have corresponding parts equal

- Range of study: A great many problems in number theory rely only on remainders when dividing by an integer Recall the division algorithm

5 Structure of the study

Chapter 1: Congruences of the first degree

Chapter 2: Problems about Congruence of the first degree

CHAPTER 1: CONGRUENCES OF THE FIRST DEGREE

1.1 Basic concepts

Our immediate problem is the study of the congruences of the general form:

𝑓(𝑥) ≡ 0(𝑚𝑜𝑑 𝑚); 𝑓(𝑥) = 𝑎𝑥𝑛+ 𝑎1𝑥𝑛−1+ ⋯ + 𝑎𝑛 (1)

• If 𝑎 is not divisible by 𝑚, then 𝑛 is said to be the degree of the congruence

• Solving a congruence means finding all the value of 𝑥 which satisfy it

• Two congruences which are satisfied by the same values of 𝑥 are said to be equivalent

If the congruence (1) 𝑓(𝑥) ≡ 0(𝑚𝑜𝑑 𝑚); 𝑓(𝑥) = 𝑎𝑥𝑛+ 𝑎1𝑥𝑛−1+ ⋯ + 𝑎𝑛 is satisfied by some 𝑥 = 𝑥1, then this congruence will also be satisfied by all numbers which are congruent to 𝑥1 modulo 𝑚:

𝑥 ≡ 𝑥1(𝑚𝑜𝑑 𝑚)

• This whole class of numbers is considered to be one solution

In accordance with this convention, congruence (1) has as many solutions as residues of a complete system satisfying it

Definition 1.1.1: Let 𝑛 be a positive integer (the modulus) We say that two integers

a, b are congruent mod n, which is written as 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), 𝑖𝑓 𝑛|𝑏 − 𝑎

Example 1.1.1:

1 If a and b are arbitrary integers, 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 1), since 1 divides every integer

and in particular it divides 𝑏 − 𝑎

2 For 𝑛 = 2, two integers 𝑎 and 𝑏 are congruent 𝑚𝑜𝑑 2 if and only if their difference 𝑏 − 𝑎 is even This happens exactly when 𝑎 and 𝑏 are both even or they are both odd

3 Something similar happens for 𝑛 = 3 Every integer has remainder 0, 1 𝑜𝑟 2 when divided by 3, and it is easy to check that 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 3) if and only if a and b have the same remainder when divided by 3 In fact, this generalizes:

As we have seen, given integers 𝑛 > 0 and a, there exist unique integers 𝑞, 𝑟 with 0 ≤ 𝑟 ≤ 𝑛 − 1, such that 𝑎 = 𝑛𝑞 + 𝑟 Here, 𝑟 is the remainder when you divide 𝑎 by 𝑛 With this said, we have the following alternate way to describe congruences

Proposition 1.1.1: Two integers 𝑎, 𝑏 are congruent mod 𝑛 if and only if they have the same remainder when divided by 𝑛

Proof First suppose that 𝑎, 𝑏 are congruent 𝑚𝑜𝑑 𝑛 Thus, 𝑏 − 𝑎 = 𝑛𝑘 for some

integer k, so that 𝑏 = 𝑎 + 𝑛𝑘 Now long division with remainder says that 𝑎 = 𝑛𝑞 +

𝑟, 𝑤𝑖𝑡ℎ 0 ≤ 𝑟 ≤ 𝑛 − 1 Hence, 𝑏 = 𝑎 + 𝑛𝑘 = 𝑛𝑞 + 𝑟 + 𝑛𝑘 = 𝑛𝑞 + 𝑛𝑘 +

𝑟 = 𝑛(𝑞 + 𝑘) + 𝑟, 𝑤𝑖𝑡ℎ 0 ≤ 𝑟 ≤ 𝑛 − 1 Thus, we have written b as a multiple of n,

namely 𝑛(𝑞 + 𝑘), plus 𝑟, with 0 ≤ 𝑟 ≤ 𝑛 − 1 By the uniqueness of long division with remainder, r is the remainder when we divide n into b So a and b have the same remainder when divided by n

Conversely, suppose that a and b have the same remainder when divided by n By

definition, 𝑎 = 𝑛𝑞1 + 𝑟 𝑎𝑛𝑑 𝑏 = 𝑛𝑞2 + 𝑟 for some integers 𝑞1, 𝑞2 Then 𝑏 − 𝑎 = 𝑛𝑞2 + 𝑟 − (𝑛𝑞1 + 𝑟) = 𝑛𝑞2 − 𝑛𝑞1 = 𝑛(𝑞2 − 𝑞1) Thus b − a is a multiple of n

We are used to seeing the integers grouped into even and odd integers Likewise,

we can group integers according to their remainders when divided by 3, or by n In general, we call the set of all integers congruent to a given integer a mod n a congruence class mod n It is easy to see that the number of congruence classes mod n is n, and that they are described by the set of possible remainders 0, 1, , n − 1 We will say a little

more about this in the next section

Let us conclude this section by saying a few words about why congruences are a good thing to study One answer is that they describe cyclical phenomena: days of the week, hours of the day, dates of the year if there are no leap years or leap centuries, It is important to have a kind of mathematics to describe such phenomena A second answer is that we might want to study certain complicated equations in integers For example, we might want to show that there are no interesting integer solutions to the equation 𝑥𝑘 + 𝑦𝑘 = 𝑧𝑘 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑘 > 2 Of course, this is a very hard equation to study! We might try to look at the easier equation 𝑥𝑘 + 𝑦𝑘 ≡ 𝑧𝑘 (𝑚𝑜𝑑 𝑛) for various

n For any given n, there are really only finitely many x, y, z to check, so that the

existence question for solutions is much easier to decide The existence or nonexistence of solutions to the congruence equation, and more generally the structure of all of the solutions, might give us some clues as to whether the original equation in integers has a solution At the end of the seminar, we will try to look at simpler examples of this idea

Proposition 1.1.2: Let n be a positive integer

1 For all 𝑎 ∈ 𝑍, 𝑎 ≡ 𝑎 (𝑚𝑜𝑑 𝑛)

2 For all 𝑎, 𝑏 ∈ 𝑍, 𝑖𝑓 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), 𝑡ℎ𝑒𝑛 𝑏 ≡ 𝑎 (𝑚𝑜𝑑 𝑛)

3 For all 𝑎, 𝑏, 𝑐 ∈ 𝑍, 𝑖𝑓 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛)𝑎𝑛𝑑 𝑏 ≡ 𝑐 (𝑚𝑜𝑑 𝑛), 𝑡ℎ𝑒𝑛 𝑎 ≡ 𝑐 (𝑚𝑜𝑑 𝑛)

These properties look like the usual properties of equality, and they are given the same names (reflexive, symmetric, transitive) Any relationship between two integers (or elements of a more general set) which satisfies all three properties is called an equivalence relation

It is easy to prove Proposition 1.1.2 For example, to see (1), for every integer a, a − a = 0, and n · 0 = 0, so 𝑛|𝑎 − 𝑎 and hence by definition 𝑎 ≡ 𝑎 (𝑚𝑜𝑑 𝑛) As for (2),

𝑛|𝑏 − 𝑎 if and only if 𝑛|𝑎 − 𝑏, 𝑠𝑜 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) if and only if 𝑏 ≡ 𝑎 (𝑚𝑜𝑑 𝑛) We leave (3) as an exercise

A typical example of how to apply these properties is as follows: given an r with

0 ≤ 𝑟 ≤ 𝑛 − 1, 𝑖𝑓 𝑎 ≡ 𝑟 (𝑚𝑜𝑑 𝑛) and 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛), then 𝑏 ≡ 𝑟 (𝑚𝑜𝑑 𝑛) as well

The second set of properties we use about congruences is that we can add and multiply them in a consistent way The following generalizes the fact that the sum of two odd numbers or two even numbers is always even, that the sum of an odd and an even number is odd, the product of two odd numbers is always odd and the product of two numbers, one of which is even, is always even

Proposition 1.1.3 Let n be a positive integer Suppose that 𝑎1 ≡ 𝑎2 (𝑚𝑜𝑑 𝑛) and that 𝑏1 ≡ 𝑏2 (𝑚𝑜𝑑 𝑛) Then 𝑎1 + 𝑏1 ≡ 𝑎2 + 𝑏2 (𝑚𝑜𝑑 𝑛) and 𝑎1𝑏1 ≡ 𝑎2𝑏2 (𝑚𝑜𝑑 𝑛)

Proof We shall just prove the second statement and leave the first as an exercise

By assumption, 𝑛|𝑎2 − 𝑎1 𝑎𝑛𝑑 𝑛|𝑏2 − 𝑏1 Write 𝑎2 − 𝑎1 = 𝑛𝑘1 and 𝑏2 − 𝑏1 = 𝑛𝑘2 Thus 𝑎2 = 𝑎1 + 𝑛𝑘1 and 𝑏2 = 𝑏1 + 𝑛𝑘2 Hence

We can view the algebraic operations of + and · as operations on the finite set of congruence classes mod n; this is an example of a finite algebraic system We will say

more about this in a minute, but first let us record some of the usual properties of congruence addition and multiplication We will not verify all of these properties; they follow immediately from the usual properties of addition and multiplication of integers The first set of properties have to do with addition:

Proposition 1.1.3 Let n be a positive integer

(i) (Associativity of addition) For all 𝑎, 𝑏, 𝑐 ∈ 𝑍,

Proposition 1.1.4 Let n be a positive integer

(i) (Associativity of multiplication) For all 𝑎, 𝑏, 𝑐 ∈ 𝑍,

Notice that we do not speak about multiplicative inverses or cancellation, and in fact we shall see that cancellation is not always possible Before that, though, we need the following property linking addition and multiplication:

Proposition 1.1.5 (Multiplication distributes over addition) Let 𝑛 be a positive integer For all 𝑎, 𝑏, 𝑐 ∈ 𝑍,

𝑎 · (𝑏 + 𝑐) ≡ 𝑎𝑏 + 𝑎𝑐 (𝑚𝑜𝑑 𝑛)

In the usual way, we always have 𝑎 · 0 ≡ 0 (𝑚𝑜𝑑 𝑛) for every 𝑎 (because 𝑎 · 0 = 𝑎 · (0 + 0) = 𝑎 · 0 + 𝑎 · 0 But notice that, if for example we take 𝑛 = 6, 2 · 3 ≡ 0 ≡ 2 · 0 (𝑚𝑜𝑑 6), although 3 is not congruent to 0 mod 6, so that we

cannot just cancel off the nonzero factor 2 For another example with 𝑛 = 9, 3 · 5 ≡ 6 ≡ 3 · 2 (𝑚𝑜𝑑 9), but we cannot cancel off the 3 to get 6 ≡ 2 (𝑚𝑜𝑑 9) Note that, if a has a multiplicative inverse mod 𝑛, i.e if there exists an 𝑥 such that 𝑎𝑥 ≡ 1 (𝑚𝑜𝑑 𝑛), then we can cancel off multiplication by 𝑎, since if 𝑎𝑏 ≡ 𝑎𝑐 (𝑚𝑜𝑑 𝑛), then multiply by 𝑥 to get

𝑥(𝑎𝑏) ≡ 𝑥(𝑎𝑐)(𝑚𝑜𝑑 𝑛); (𝑥𝑎)𝑏 ≡ (𝑥𝑎)𝑐 (𝑚𝑜𝑑 𝑛);

1 · 𝑏 ≡ 1 · 𝑐 (𝑚𝑜𝑑 𝑛) Thus 𝑏 ≡ 𝑐 (𝑚𝑜𝑑 𝑛)

Let us collect more information on congruence addition and multiplication We will work out the example 𝑛 = 6 We just write down the possible remainders 𝑚𝑜𝑑 6 in the following table for addition So the meaning of the entry corresponding to the row labeled by 3 and the column labeled by 4 is that, if 𝑎 ≡ 3 (𝑚𝑜𝑑 6) and 𝑏 ≡ 4 (𝑚𝑜𝑑 6), then 𝑎 + 𝑏 ≡ 1 (𝑚𝑜𝑑 6)

1.2 Equations in congruence

Note that the above was long and tedious enough, even though we skipped a few steps! (Which ones?) Of course, most of you would do this in one step in your head Multiplicative equations are another story: they need not always have a solution, and if they do, the solution need not be unique! For example, the equation 2𝑥 ≡ 3 (𝑚𝑜𝑑 8) has no solution: if it did, 8 would divide 2𝑥 − 3, which is always an odd number On the other hand, the equation 2𝑥 ≡ 4 (𝑚𝑜𝑑 8) has two solutions 𝑚𝑜𝑑 8: 𝑥 ≡ 2 (𝑚𝑜𝑑 8) and 𝑥 ≡ 6 (𝑚𝑜𝑑 8) On the other hand, the equation 5𝑥 ≡ 3 (𝑚𝑜𝑑 7) has exactly one solution 𝑚𝑜𝑑 7: 𝑥 ≡ 2 (𝑚𝑜𝑑 7) So we need some criterion to decide when an equation of the form 𝑎𝑥 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) has a solution, and, if so, how many We will just discuss the existence question here and leave the problem of deciding how many solutions there are to the exercises Fortunately, the work has already been done for us in the very first lecture

Corollary 1 The equation 𝑎𝑥 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) has a solution if and only if 𝑑 =

𝑔𝑐𝑑(𝑎, 𝑛) 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑏

Proof The proof follows by writing out the definitions carefully and seeing what

they say The equation 𝑎𝑥 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) has a solution if and only if 𝑛 divides 𝑎𝑥 − 𝑏

for some x if and only if there exist integers 𝑥 and k such that 𝑎𝑥 − 𝑏 = 𝑛𝑘, if and only if 𝑏 = 𝑎𝑥 + 𝑛(−𝑘) Set 𝑦 = −𝑘; clearly 𝑏 = 𝑎𝑥 + 𝑛(−𝑘) for some integers 𝑥 and k if and only if 𝑏 = 𝑎𝑥 + 𝑛𝑦 for some integers 𝑥 and 𝑦 Says that the equation 𝑏 = 𝑎𝑥 + 𝑛𝑦 has a solution in integers 𝑥 and 𝑦 if and only if 𝑑 = 𝑔𝑐𝑑(𝑎, 𝑛) divides b Running through the chain of logical equivalences, we see that 𝑎𝑥 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) has a solution if and only if 𝑑 = 𝑔𝑐𝑑(𝑎, 𝑛) divides 𝑏

Let us give some easy consequences of the above The first has to do with the existence of multiplicative inverses After all, a multiplicative inverse of a mod n is just a solution to the equation 𝑎𝑥 ≡ 1 (𝑚𝑜𝑑 𝑛) So we see:

Corollary 2 There exists a multiplicative inverse for a mod n if and only if 𝑎 and 𝑛 are relatively prime, 𝑔𝑐𝑑(𝑎, 𝑛) = 1

Proof There exists a multiplicative inverse for a 𝑚𝑜𝑑 𝑛 if and only if the equation 𝑎𝑥 ≡ 1 (𝑚𝑜𝑑 𝑛) has a solution, if and only if the 𝑔𝑐𝑑 of 𝑎 and 𝑛 divides 1, if and only if the 𝑔𝑐𝑑 of 𝑎 and 𝑛 is equal to 1

An 𝑥 such that 𝑎𝑥 ≡ 1 (𝑚𝑜𝑑 𝑛) is usually written as 𝑎−1 , with the understanding that this is not the same as the rational number 1

𝑎, and that the answer will depend on n For example, 2−1 (𝑚𝑜𝑑 11) = 6, 𝑏𝑢𝑡 2−1 (𝑚𝑜𝑑 17) = 9

The next corollary says that we can cancel off relatively prime factors (In fact, this is an if and only if statement in a certain sense; see the exercises.)

Corollary 3 Suppose that 𝑎 and 𝑛 are relatively prime, and that 𝑎𝑏 ≡ 𝑎𝑐 (𝑚𝑜𝑑 𝑛) Then 𝑏 ≡ 𝑐 (𝑚𝑜𝑑 𝑛)

Proof If 𝑎 and 𝑛 are relatively prime and 𝑎𝑏 ≡ 𝑎𝑐 (𝑚𝑜𝑑 𝑛), find an x such 𝑎𝑥 ≡ 1 (𝑚𝑜𝑑 𝑛), i.e find 𝑎−1 Then multiplying the equality 𝑎𝑏 ≡ 𝑎𝑐 (𝑚𝑜𝑑 𝑛) gives 𝑏 ≡ 𝑥𝑎𝑏 ≡ 𝑥𝑎𝑐 ≡ 𝑐 (𝑚𝑜𝑑 𝑛)

We will use the next corollary later (It could have been proven easily in the first lecture, with a slightly different proof.)

Corollary 4 Let 𝑛 be a positive integer Then 𝑎1 and 𝑎2 are both relatively prime to 𝑛 if and only if 𝑎1𝑎2 is relatively prime to 𝑛

Proof Suppose that 𝑎1 and 𝑎2 are both relatively prime to 𝑛 Find 𝑥1 such that 𝑎1𝑥1 ≡ 1 (𝑚𝑜𝑑 𝑛) and find 𝑥2 such that 𝑎2𝑥2 ≡ 1 (𝑚𝑜𝑑 𝑛) Then (𝑎1𝑎2)(𝑥1𝑥2) ≡

1 (𝑚𝑜𝑑 𝑛) In other words, 𝑎1𝑎2 has the multiplicative inverse 𝑥1𝑥2 𝑚𝑜𝑑 𝑛 So, by the other direction, 𝑎1𝑎2 and 𝑛 are relatively prime

Conversely, if 𝑎1𝑎2 is relatively prime to 𝑛, then there exists 𝑥 such that (𝑎1𝑎2)𝑥 ≡ 1 (𝑚𝑜𝑑 𝑛) But then 𝑎1(𝑎2𝑥) ≡ 1 (𝑚𝑜𝑑 𝑛), so that 𝑎2𝑥 is a multiplicative inverse for 𝑎1 𝑚𝑜𝑑 𝑛 Thus 𝑎1 and 𝑛 are relatively prime Likewise, 𝑎1𝑥 is a multiplicative inverse for 𝑎2 𝑚𝑜𝑑 𝑛, so that 𝑎2 and 𝑛 are relatively prime

Note that congruences to a prime modulus look especially nice from the point of view of multiplicative inverses The reason is that, if 𝑝 is a prime number, then 𝑝 and 𝑎 are not relatively prime if and only if 𝑝 divides 𝑎

Corollary 5 Let 𝑝 be a prime number If a is not congruent to zero 𝑚𝑜𝑑 𝑝, then there exists a multiplicative inverse for 𝑎 𝑚𝑜𝑑 𝑝

The above says that, when working with congruences mod a prime number 𝑝, we can add, subtract, multiply, and divide by all nonzero numbers So in the sense

𝑚𝑜𝑑 𝑝 arithmetic is like arithmetic with the rational numbers

1.3 Congruences in the first degree

A great many problems in number theory rely only on remainders when dividing by an integer Recall the division algorithm: given 𝑎 ∈ 𝑍 and 𝑛 ∈ 𝑁 there exist unique 𝑞, 𝑟 ∈ 𝑍 such that

𝑎 = 𝑞𝑛 + 𝑟, 0 ≤ 𝑟 < 𝑛 (∗)

It is to the remainder r that we now turn our attention

Definition 1.3.1: Congruences and 𝒁𝒏

For each 𝑛 ∈ 𝑁, the set 𝑍𝑛 = {0, 1, , 𝑛 − 1} comprises the residues modulo

n Integers a, b are said to be congruent modulo n if they have the same residue: we

is satisfied by two numbers 𝑥 = 2 and 𝑥 = 4 among the numbers 0, 1, 2, 3, 4, 5, 6 of a complete residue system 𝑚𝑜𝑑𝑢𝑙𝑜 7 since:

when 𝑥 = 2: 25+ 2 + 1 = 35 ⋮ 7 when 𝑥 = 4: 45+ 4 + 1 = 1029 ⋮ 7

Therefore the above congruence has the two solutions: 𝑥 ≡ 2(𝑚𝑜𝑑 7), 𝑥 ≡ 4(𝑚𝑜𝑑 7)