Lesson 3 Integers - Kuina-chan Mathematics

March 20, 2026

Kuina-chan

“Kuina-chan’s Math” Lesson 3 explains integers, including negative numbers! It is assumed you have read Lesson 1.

Lesson 2 proved “

” using the axioms of sets, natural numbers, and addition. However, even without bringing up these axioms, we are convinced that “

”. So, we will stop proving things like “

$\times$

” and “

” in the manner of the previous lesson, accepting that they can be proved if one is so inclined, and from now on focus on “things we don’t really know if they hold true”.

1.Integers

In Lesson 2, we expressed natural numbers as the set “ $\mathbb{N}$

$\{$

$\dots$ $\}$ ”, but numbers with a minus sign attached to them (except

) are called “integers”. That is, if the set of all integers is $\mathbb{Z}$ , then “ $\mathbb{Z}$ $\{$ $\dots$ $\dots$ $\}$ ”.

Numbers greater than

are called “positive” numbers, and numbers less than

are called “negative” numbers.

is neither.

As you know, for any two integers

, we can perform addition “

”, subtraction “

”, and multiplication “

$\times$

”. “

$\times$

” is sometimes written as “ $\cdot$ ”, or often the multiplication symbol is omitted and written as “”. In this article, we will write it that way from now on.

1.1Exponentiation

For an integer

and an integer

greater than or equal to

, “ multiplied times” is written as “ $^{b}$ ” and is called “exponentiation”. For example, “

$^{3}$ ” is “

$\cdot$

”, which is

. “

$^{4}$ ” is “

$\cdot$

”, which is

However, for any number

that is not

, we define “

$^{0}$

”. For example, “

$^{0}$

” and “

$^{0}$

”.

Note

If you look at “2⁵ = 32”, “2⁴ = 16”, “2³ = 8”, “2² = 4”, “2¹ = 2”, the result is halved each time, so you can see that it is natural to think “2⁰ = 1”.

“

$^{0}$ ” is sometimes defined as “

” for convenience, but for various reasons it is often left undefined.

Note

One reason why “0⁰” is usually undefined is that while looking at “3⁰ = 1”, “2⁰ = 1”, “1⁰ = 1” makes it natural to think “0⁰ = 1”, looking at “0³ = 0”, “0² = 0”, “0¹ = 0” makes it natural to think “0⁰ = 0”, leading to a contradiction.

The following laws hold for exponentiation.

(1) is clear because multiplying “

$\cdot$

$\cdot$ $\dots$ $\cdot$

(

times)” and “

$\cdot$

$\cdot$ $\dots$ $\cdot$

(

times)” results in “

$\cdot$

$\cdot$ $\dots$ $\cdot$

(

times in total)”.

(2) involves division, which reduces the number of

’s, so the count of

becomes subtraction.

(3) means “

$\cdot$

$\cdot$ $\dots$ $\cdot$

(

times)” itself is repeated

times, resulting in “

$\cdot$

$\cdot$ $\dots$ $\cdot$

(

$\cdot$

times)”.

(4) means “

$\cdot$

$\cdot$ $\dots$ $\cdot$

$\cdot$

(

times each for

and

)”, so rearranging the order gives “

$\cdot$

$\cdot$ $\dots$ $\cdot$

$\cdot$

$\cdot$ $\dots$ $\cdot$

(

times each for

and

)”.

1.2Absolute Value

How far an integer

is from

is called the “absolute value” of

, denoted as “”. For example, the absolute value of

is “

”, and the absolute value of

is “

”.

Absolute value can be thought of as “leaving positive numbers as they are, and removing the minus sign from negative numbers”.

More strictly defined, it is as follows.

For example, if

, since

, then

2.Properties of Integers

Now, let’s explain various properties of integers.

2.1Quotient and Remainder

Division of two integers (

) may result in a value that is not an integer. Therefore, we define “quotient” and “remainder” where the calculation results remain integers.

When performing “

”, the “quotient” is the number of items per person when items are distributed among people. The “remainder” is the number of items left over that couldn’t be distributed. For example, for “

”, the quotient is

and the remainder is

“The quotient of

being

and remainder

” means “when items are distributed among people, each gets items and item remains”, which can be rephrased as “there are people each with items, and combining those with the remaining item gives items”. This can be written as “

$\cdot$

”. In other words, “quotient and remainder of ” are defined as numbers satisfying “ $\cdot$ ”.

For example, considering “

”, the quotient is

and remainder is

. Substituting

with

, quotient

with

, and remainder

with

into the above equation gives “ $\cdot$ and

$\leq$

”, which indeed satisfies the mathematical expression.

In the above expression, the quotient and remainder when

are undefined. That is, “

” etc. are undefined.

2.2Divisibility, Divisors, and Multiples

If the remainder of is , we say “

divides

” (or

is divisible by

). For example, “

” has a remainder of

, so

divides

. Also, “

” has a remainder of

, so

divides

When

divides

is called a “divisor” of

, and

is called a “multiple” of

. For example, since

divides

is a divisor of

, and

is a multiple of

Listing the divisors of

in ascending order gives “

”. The multiples of

are “ $\dots$

$\dots$ ”, which is the set of all even numbers.

Since

and

divide all integers, multiples of

and

are all integers. All integers except

divide

, so the divisors of

are all integers except

2.3Common Divisors and Common Multiples

Now let’s consider common divisors and multiples of two or more integers.

A divisor common to both

and

is called a “common divisor” of

and

. In other words, if

divides

and

divides

, the integer

is called a “common divisor” of

and

. For example,

divides

and

divides

, so

is one of the common divisors of

and

A multiple common to both

and

is called a “common multiple” of

and

. In other words, if

divides

and

divides

, the integer

is called a “common multiple” of

and

. For example,

divides

and

divides

, so

is one of the common multiples of

and

For three or more numbers, common divisors and common multiples can be defined similarly.

2.4Greatest Common Divisor and Least Common Multiple

The largest of the common divisors of

and

is called the “greatest common divisor” of

and

, often denoted as “ $\rm{g}$ $\rm{c}$ $\rm{d}$

”. The smallest of the positive common multiples of

and

is called the “least common multiple” of

and

, often denoted as “ $\rm{l}$ $\rm{c}$ $\rm{m}$

”.

Note

gcd stands for “greatest common divisor”, and lcm stands for “least common multiple”.

For example, the divisors of

are “

”, and the divisors of

are “

”. The common divisors of

and

are the shared “

”, and the greatest common divisor is the largest among them, so $\rm{g}$ $\rm{c}$ $\rm{d}$

Also, the positive multiples of

are “

$\dots$ ”, and the positive multiples of

are “

$\dots$ ”. The positive common multiples of

and

are the shared “

$\dots$ ”, and the least common multiple is the smallest among them, so $\rm{l}$ $\rm{c}$ $\rm{m}$

For positive integers

, there is a law that “ $\cdot$ $\rm{g}$ $\rm{c}$ $\rm{d}$ $\cdot$ $\rm{l}$ $\rm{c}$ $\rm{m}$ ”. For example, since “ $\rm{g}$ $\rm{c}$ $\rm{d}$

” and “ $\rm{l}$ $\rm{c}$ $\rm{m}$

”, substituting into “

$\cdot$

$\rm{g}$ $\rm{c}$ $\rm{d}$

$\cdot$ $\rm{l}$ $\rm{c}$ $\rm{m}$

” gives “

$\cdot$

”, resulting in “

”, which holds true. Using this, if you know either the greatest common divisor or the least common multiple, you can easily calculate the other.

2.5Euclidean Algorithm

Finding the greatest common divisor directly takes time, but using the “Euclidean algorithm” described below makes it faster.

For example, finding the greatest common divisor of and using the Euclidean algorithm is as follows.

Generally, repeating division is easier than listing common divisors, making this method convenient.

3.Prime Numbers

An integer

greater than or equal to

whose positive divisors are only and is called a “prime number”. For example,

is a prime number because its positive divisors are only

and

is not a prime number because it has

as a divisor in addition to

and

In other words, a prime number is an integer greater than or equal to

that “cannot be divided by any positive integer other than ‘1 and itself’”. Integers greater than or equal to

that are not prime numbers are called “composite numbers”.

Listing prime numbers in ascending order gives “

$\dots$ ”. There are infinitely many prime numbers. The appearance of prime numbers seems irregular, and research to capture their rules has continued from ancient times to the present.

Prime numbers can be obtained using a method called the “Sieve of Eratosthenes”. This method uses the fact that “among integers greater than or equal to

, those that are not multiples of other prime numbers are prime numbers”, and is performed as follows.

3.1Prime Factorization

All positive integers can be expressed as a product of prime numbers. For example, “

$\cdot$

”, “

$\cdot$

”, “

$\cdot$

”, etc. Expressing a positive integer as a product of prime numbers in this way is called “prime factorization”.

Each prime number that appears when prime factorized is called a “prime factor”. For example, since “

$\cdot$

”, the prime factors of

are

and

Any positive integer can always be prime factorized, and the pattern is limited to one way if the order of multiplication is ignored. For example, using exponentiation, “

$^{0}$

$^{0}$ $\dots$ ”, “

$^{1}$

$^{0}$

$^{0}$ $\dots$ ”, “

$^{0}$

$^{1}$

$^{0}$ $\dots$ ”, “

$^{2}$

$^{0}$

$^{0}$ $\dots$ ”, “

$^{0}$

$^{1}$ $\dots$ ”, “

$^{1}$

$^{0}$ $\dots$ ”. This property is called the “Unique Factorization Theorem” (or uniqueness of prime factorization) and is useful for proving other theorems.

The reason

is not included in prime numbers is that if

were included, “

$^{0}$

$^{1}$ $\dots$

$^{1}$

$^{1}$ $\dots$

$^{2}$

$^{1}$ $\dots$

$^{3}$

$^{1}$ $\dots$ ”, and the uniqueness of prime factorization would no longer hold.

3.2Coprime

When two integers

and

have no common divisors other than

and

, that is, when $\rm{g}$ $\rm{c}$ $\rm{d}$

and

are said to be “coprime” (or relatively prime). For example, since $\rm{g}$ $\rm{c}$ $\rm{d}$

and

are coprime.

Saying that positive integers

and

are “coprime” is equivalent to saying that

and

have “no common prime factors”. For example, from

$\cdot$

and

$\cdot$

and

do not contain common prime factors, so they can be said to be coprime.

4.Modular Arithmetic

Now, returning to the topic of remainders in division, “the remainder when

is divided by

is ”, and “the remainder when

is divided by

is also ”, so they match. This can be said as “in the world of remainders when integers are divided by ,

holds true”. When the remainders divided by

match in this way, we say “ and are congruent modulo ” and write “ $\equiv$ $\rm{m}$ $\rm{o}$ $\rm{d}$ ”.

Generally, for a positive integer

, when the remainders of

and

match, we say “ and are congruent modulo ” and write “ $\equiv$ $\rm{m}$ $\rm{o}$ $\rm{d}$ ”. When they do not match, we write “ $\not\equiv$ $\rm{m}$ $\rm{o}$ $\rm{d}$ ”. An expression written this way is called a “congruence equation” (or modular arithmetic).

For example, “the remainder of

divided by

” is the same as “the remainder of

divided by

”, so “

$\equiv$

$\rm{m}$ $\rm{o}$ $\rm{d}$

”. On the other hand, “the remainder of

divided by

” is different from “the remainder of

divided by

”, so “

$\not\equiv$

$\rm{m}$ $\rm{o}$ $\rm{d}$

”.

Congruence equations have the property that they hold even if the same number is added, subtracted, or multiplied to both sides.

For example, since “

$\equiv$

$\rm{m}$ $\rm{o}$ $\rm{d}$

” holds, multiplying both sides by

gives “

$\equiv$

$\rm{m}$ $\rm{o}$ $\rm{d}$

”, which also holds.

5.Diophantine Equations

Finally, let’s challenge a specific problem applying the properties of integers introduced so far. It is a problem called “Diophantine equation”.

An “equation” is a problem to find the value of a variable that satisfies an equality, such as “Find

satisfying

”. The value of the variable for which the equality holds is called the “solution” of the equation.

Among equations, “Diophantine equations” refer to those where the equation has infinitely many solutions (over real numbers, but typically integer solutions are sought). For example, “Find the combination of

and

satisfying

”. In this case, “

” and “

” are solutions.

In this way, Diophantine equations often have infinitely many solutions, but by adding conditions, the number of solutions can become finite. Let’s look at a problem that can be solved like a puzzle by using such conditions.

5.1Problem

Let’s challenge the following specific Diophantine equation problem.

5.2Solution Method

First, let’s construct the Diophantine equation. Let the

-digit integer

have digits

from the top. For example, if

, then

. Then

can be expressed as

Since reversing it results in

times the original number, the following equation is formed.

The left side is

times

, and the right side is the reversed number.

As it is, this expression contains 4 variables and is a Diophantine equation with potentially many solutions, so let’s narrow down the solutions using various conditions.

5.3Finding the value of a

First, if

would be

digits or less, so it must be

. Also, if

$\geq$

, multiplying by

would result in

digits or more, so it must be

. That is, is either or .

Here, if we assume

, the equation becomes “

$\cdot$

”, and the ones digit on the right side is “1”. The left side is an integer multiplied by

, but there is no integer that becomes 1 in the ones digit when multiplied by 4 (it must be even), so the left and right sides never match. In other words, it is clear that no solution exists when

. Therefore, if a solution exists, it is only when .

5.4Finding the value of d

Substituting

, the equation becomes “

$\cdot$

”. Here, the ones digit on the right side is “

”. The ones digit of an integer multiplied by

becoming

only happens for “

$\cdot$

” and “

$\cdot$

”, so , which is the ones digit of , is either or .

, the equation is “

$\cdot$

”, but organizing this gives “

”. Substituting any value from

into

results in

being a negative number, so

$\neq$

. Therefore, if a solution exists, it is only when .

5.5Finding the values of b and c

Substituting

, the equation becomes “

$\cdot$

”. Transforming this gives “

”. For “

” to be an integer, trying values from

for

shows that is the only solution.

Substituting

into

, we get

$\cdot$

, so .

Therefore, from

, we get . Calculating

$\cdot$

gives

, confirming that “multiplying by

results in the reverse order of the original number”.

In this lesson, we introduced basic properties of integers. Next time, we will explain “real numbers” and “functions” and “mappings”, which are important for handling these numbers!

Lesson 3Integers

1.Integers

1.1Exponentiation

1.2Absolute Value

2.Properties of Integers

2.1Quotient and Remainder

2.2Divisibility, Divisors, and Multiples

2.3Common Divisors and Common Multiples

2.4Greatest Common Divisor and Least Common Multiple

2.5Euclidean Algorithm

3.Prime Numbers

3.1Prime Factorization

3.2Coprime

4.Modular Arithmetic

5.Diophantine Equations

5.1Problem

5.2Solution Method

5.3Finding the value of a

5.4Finding the value of d

5.5Finding the values of b and c

Lesson 3
Integers