Study Guide on Divisibility and the Properties of Natural and Integer Numbers

Properties of Natural Numbers and Integers

Within the set of Natural Numbers ($\mathbb{N}$), covering specifically non-zero naturals, every number $n$ possesses specific relational properties. For any natural number $n$, where $n \neq 0$, there exists an antecessor defined as $n - 1$ and a successor defined as $n + 1$. These represent the immediate values preceding and following the number in the natural sequence.

Parity is another fundamental characteristic of natural numbers. An even number is any value that can be expressed in the form $2n$, where $n$ is a natural number. Conversely, an odd number is defined by the mathematical expression $2n + 1$, where $n$ is a natural number.

In the set of Integers ($\mathbb{Z}$), the concept of an opposite or symmetric number is introduced. The opposite of a number is that which maintains the exact same distance from zero as the number itself. For instance, the numbers $5$ and $-5$ are opposites, as are $-18$ and $18$. This leads to the definition of the Module or Absolute Value, which represents the literal distance between a specific number and zero. This is denoted by vertical bars; for example, $|-5| = 5$ and $|8| = 8$.

Division in the Set of Natural Numbers

The process of dividing a natural number $D$ by a natural number $d$, where $d > 0$, requires the determination of two specific natural numbers: the quotient ($q$) and the remainder ($r$). This relationship is governed by the Dividend Identity, expressed as $D = d \times q + r$. A critical constraint of this operation is that the remainder must be less than the divisor and at least zero, such that $0 \leq r < d$.

This mathematical framework is formally known as the Division Algorithm or Euclidean Division. In this system, the four participating elements are strictly defined: $D$ represents the dividend, $d$ serves as the divisor, $q$ is the quotient, and $r$ is the remainder.

Concepts of Divisibility, Divisors, and Multiples

A division is classified as exact when the remainder is zero ($r = 0$). In such instances, we state that $D$ is divisible by $d$ and express the relationship as $D = d \times q$. The statement "$D$ is divisible by $d$" is logically equivalent to saying that "$d$ divides $D$", which is mathematically denoted as $d|D$.

The set of Divisors of a number $a$, denoted as $D(a)$, is the collection of all integers that can divide $a$ without leaving a remainder. In contrast, the set of Multiples of a number $a$, denoted as $M(a)$, comprises the integers that $a$ can divide.

Criteria for Divisibility

To determine if a number is divisible by another without performing a full long division, specialized rules or criteria are used for specific divisors:

Divisibility by 1: This is considered the most trivial criterion, as virtually every integer is divisible by 1.

Divisibility by 2: An integer is divisible by 2 if it is an even number. This includes any number whose final digit is $0$, $2$, $4$, $6$, or $8$.

Divisibility by 6: For a number to be divisible by 6, it must satisfy the divisibility requirements for both 2 and 3 simultaneously. This means the number must be even and the sum of its digits must be a multiple of 3.

Divisibility by 8: A number is divisible by 8 if it ends in $000$ or if its three most significant final digits form a number that is itself divisible by 8.

Divisibility by 9: If the sum of all individual digits within a certain number is divisible by 9, then the entire number is divisible by 9.

Divisibility by 10: Any number that terminates with the digit $0$ is divisible by 10.

Prime and Composite Numbers

There are two primary ways to define Prime Numbers. Definition 1 states that a prime number is an element of a set where the numbers possess exactly two natural divisors (specifically 1 and the number itself). For example, $2$ is prime (divisors are 1 and 2), and $7$ is prime (divisors are 1 and 7). Under this definition, $6$ is not prime because it has four natural divisors: 1, 2, 3, and 6.

Definition 2 focuses on integers, stating that a prime number is part of the set of numbers possessing exactly four integer divisors. For instance, $2$ is prime because its integer divisors are $\pm 1$ and $\pm 2$. Similarly, $7$ is prime as its integer divisors are $\pm 1$ and $\pm 7$. Again, $6$ fails this criteria as it possesses eight integer divisors ($\pm 1$, $\pm 2$, $\pm 3$, and $\pm 6$).

A number that has more than four integer divisors is referred to as a Composite Number or a Multiple Number. The number $6$ serves as a primary example of a composite number. Important observations regarding primes include the fact that the number $1$ is not considered prime, and the number $2$ stands as the only even prime number.

The Fundamental Theorem of Arithmetic and Factorization

The Fundamental Theorem of Arithmetic dictates that every natural number greater than 1 is either a prime number itself or can be uniquely expressed as a product of prime numbers. This unique representation is the basis for Numerical Factorization.

Factorization, or decomposing a number into prime factors, involves rewriting a number as a multiplication of primes. During this process, prime numbers are typically used in ascending order, adhering to the divisibility rules applicable to the term being factored. Below are provided examples of numeric factorization:

a) $24 = 2 \times 2 \times 2 \times 3$

b) $10 = 2 \times 5$

c) $52 = 2 \times 2 \times 13$

d) $112 = 2 \times 2 \times 2 \times 2 \times 7$

e) $600 = 2 \times 2 \times 2 \times 3 \times 5 \times 5$