Arithmetic

Overview

Arithmetic refers to the branch of mathematics dealing with, at its most basic level, addition, subtraction, multiplication, and division. There exist names for each term in these basic expressions:

In $a + b$ , $a$ is the augend and $b$ is the addend.
In $a - b$ , $a$ is the minuend and $b$ is the subtrahend.
In $a \times b$ , $a$ is the multiplicand and $b$ is the multiplier.
In $a \div b$ , $a$ is the dividend and $b$ is the divisor.

In a wider sense, arithmetic may also include exponentiation and logarithms.

Division Algorithm

If $m$ is a positive integer and $n$ is any integer, then there exist unique integers $q$ and $r$ such that

n = m q + r and 0 \leq r < m .

Value $q$ is called the quotient whereas value $r$ is called the remainder when $n$ is divided by $m$ . We say $m$ divides $n$ if the remainder when $n$ is divided by $m$ is $0$ . This is denoted as $m ∣ n$ .

Prime Numbers

A prime number is a natural number greater than $1$ that is not a product of two smaller natural numbers. A composite number is a natural number greater than $1$ that is not prime.

We say two positive integers $a$ and $b$ are relatively prime if their greatest common divisor is $1$ . Alternatively, we say $a$ and $b$ are coprime.

Euclid's Lemma

If an integer $n$ divides the product $a b$ of two integers, and is coprime with $a$ , then $n$ divides $b$ . More generally, if an integer $n$ divides the product $a_{1} a_{2} \dots a_{m}$ of $m$ integers, and is coprime with each of $a_{1}, a_{2}, \dots, a_{m - 1}$ , then $n$ divides $a_{m}$ .

Fundamental Theorem of Arithmetic

The fundamental theorem of arithmetic, also known as the unique factorization theorem, states that every integer greater than $1$ is uniquely represented as a product of prime numbers.