This page contains basic area of number system with some topics in elementary number theory. The readers should have a little knowledge of factors, multiples, lcm, hcf/gcd, prime numbers and composite numbers.

Euclid’s Division Lemma

Euclid’s division lemma states that for any two integers $a$ and $b$ with $b>0$, there exist unique integers $q$ and $r$ such that $a=bq+r$ and $0\le r < b.$

The integer $q$ is called the quotient of $a$ with respect to $b$ and the integer $r$ is called the remainder of $a$ with respect to $b.$

In other words, we have $\text{Dividend}=\text{Divisor}\times\text{Quotient}+\text{Remainder}.$

Euclid’s Algorithm

Euclid’s algorithm for finding the HCF of two given positive integers are given below.

1. Find the quotient and remainder of the division of the greater number by the smaller.
2. If the remainder is zero, then the divisor is the HCF.
3. Else, taking the previous remainder as the new divisor and the previous divisor as the new dividend, find the quotient and the remainder.
4. Continue the process till the remainder is zero. The last divisor is the required HCF.

Remark. Euclid’s algorithm can be extended for all non-zero integers.

Notation. The HCF of $a$ and $b$ is denoted by $\gcd(a,b)$ or simply by $(a,b).$ The LCM of $a$ and $b$ is denoted by $\mathrm{lcm}[a,b]$ or simply by $[a,b].$

Euclid’s algorithm works because of the following result.

Theorem. If $a=bq+r$, then $(a,b)=(b,r).$

Fundamental Theorem of Arithmetic

Fundamental theorem of arithmetic or unique factorisation theorem states that every composite number can be expressed as a product of primes uniquely except for the order of the factors.

The fundamental theorem of arithmetic can also be stated in the following form.

Every integer $n>1$ can be expressed uniquely in the form \[ n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k},\tag{1} \] where $p_1,p_2,\ldots, p_k$ are primes such that $p_1<p_2<\cdots<p_k$ and $a_1,a_2,\ldots,a_k$ are all positive integers.

The factorisation of the number $n$ as given by (1) is called the standard or canonical decomposition of $n.$

Field Properties of Real Numbers

The rational numbers and irrational numbers together form the system of real numbers. Some basic properties (known as the field properties) of the real numbers associated with the operations of addition and multiplication are stated below.

1. Closure under addition. The sum of two real numbers is a real number, i.e., $x+y\in\mathbb{R}$ whenever $x,y\in\mathbb{R}.$
2. Associativity of addition. $(x+y)+z=x+(y+z)\ \forall\ x,y,z\in\mathbb{R}.$
3. Commutativity of addition. $x+y=y+x\ \forall\ x,y\in\mathbb{R}.$
4. Existence of additive identity. There exists a real number 0 called the additive identity such that $x+0=x$ for every $x\in \mathbb{R}.$
5. Existence of additive inverse. For every $x\in\mathbb{R}$, there exists $-x\in\mathbb{R}$ called the additive inverse or negative of $x$ such that $x+(-x)=0.$
6. Closure under multiplication. The product of two real numbers is a real number, i.e., $xy\in\mathbb{R}$ whenever $x,y\in\mathbb{R}.$
7. Associativity of multiplication. $(xy)z=x(yz)\ \forall\ x,y,z\in\mathbb{R}.$
8. Commutativity of multiplication. $xy=y\ \forall\ x,y\in\mathbb{R}.$
9. Existence of multiplicative identity. There exists a real number 1 called the multiplicative identity such that $x\times1=x$ for every $x\in \mathbb{R}.$
10. Existence of multiplicative inverse. For every non-zero $x\in\mathbb{R}$, there exists $\dfrac{1}{x}\in\mathbb{R}$ called the multiplicative inverse of $x$ such that $x\times\dfrac{1}{x}=1.$
11. Multiplication distributes over addition. $x(y+z)=xy+xz$ for all $x,y,z\in\mathbb{R}.$

Some results which can be deduced from the the field properties of real numbers are given below.

1. Cancellation law for addition. If $x,y,z\in\mathbb{R}$ and $x+y=x+z$, then $y=z.$
2. Cancellation law for multiplication. If $x,y,z\in\mathbb{R},x\ne0$ and $xy=xz$, then $y=z.$
3. For any $x\in\mathbb{R}$, $x\cdot0=0.$
4. For any $x,y\in\mathbb{R}$, $x(-y)=-xy.$
5. For any $x,y\in\mathbb{R}$, $(-x)(-y)=xy.$
6. If $x,y\in\mathbb{R}$ and $xy=0$, then $x=0$ or $y=0.$

Absolute Value of a Real Number

The absolute value or modulus of a real number $x$, denoted by $|x|$, is defined by $$ |x| = \left\{ \begin{array}{rl} x & \text{if } x\ge0;\\ -x & \text{if } x<0. \end{array}\right. $$

It can also be defined as follows. \[ |x| = \left\{ \begin{array}{ll} 0 & \text{if } x=0;\\ \text{the greater of $x$ and $-x$} & \text{if } x\neq0. \end{array}\right. \]

Some properties of absolute values of real numbers are given below.

1. For any $x\in\mathbb{R}$, $|x|\ge0.$
2. For any $x\in\mathbb{R}$, $|-x|=|x|.$
3. For any $x,y\in\mathbb{R}$, $|xy|=|x||y|.$
4. Triangle inequality. For any $x,y\in\mathbb{R}$, $|x+y|\le|x|+|y|.$
5. For any $x,y\in\mathbb{R}$, $|x-y|\ge||x|-|y||.$
6. For $x,y,\delta\in\mathbb{R}$, where $\delta>0$, $|x-y|<\delta$ if and only if $y-\delta<x<y+\delta.$