The study of positive whole numbers or natural numbers is known as number theory or higher arithmetic. The main goal of number theory is to discover and analyse unique relationships between different kinds of numbers. This topic has its concepts originating from the ancient mathematics division. The principles of number theory are the building blocks of many concepts of different disciplines of Mathematics.

#### Types of Numbers

The number theory classifies numbers into different categories like natural numbers, complex numbers, whole numbers, etc. The set of natural numbers is further divided into these categories:

• Odd Numbers: Numbers that are not evenly divisible by 2 are called odd numbers. For example, 1,3,5,7,9,11,13,15,………
• Even Number: Numbers that are evenly divisible by 2 are called even numbers. For example, 2,4,6,8,10,12,14,16…………
• Square Numbers: When a number goes through multiplication with itself, the resultant number is known as a square number or a perfect square. For example, 2×2 = 4, 3×3 = 9, 4×4 = 16. These are some perfect squares 1,4,9,16,25,36,49,64,81……..
• Cube Numbers: When a number goes through multiplication with itself twice, the resultant number is known as a cube number or a perfect cube. For example, 2x2x2 = 8, 3x3x3 = 27, 4x4x4 = 64. These are some perfect cubes 1,8,27,64,125,216, 343, 512……….
• Prime Number: If a number has only two factors, 1 and itself, then the number is known as a prime number. For example,  2,3,5,7,11,13,17,19,23,29………
• Co-Prime Numbers: If the highest common factor between two numbers is 1, then the numbers are called co-primes. For example, 2 and 3, 3 and 4, 4 and 5, and so on.
• Composite Numbers: Whole numbers that are non-prime and have two or more than two factors are called composite numbers. 1 is neither prime nor composite in nature. For example, 4, 6, 8, 9, 10, 12, 14, 15,…………
• Modulo 4 Numbers: The numbers are called 1(modulo 4) if they leave a remainder 1 when divided by 4 and 3(modulo 4) if they leave a remainder 3 when divided by 4. For example,

1 (modulo 4) Numbers – 1, 5, 9, 13, 17, 21, 25, . . .

3 (modulo 4) Numbers – 3, 7, 11, 15, 19, 23, 27, . . .

• Triangular Numbers: A number is called a triangular number if that number of pebbles can be arranged in a triangular pattern with one pebble at the top, two pebbles in the second row and so on. For example, 3,6,10,15,21,28,36……
• Fibonacci Numbers: The set of numbers in which the sum of any two adjacent numbers is the third adjacent number towards the ascending side is called fibonacci numbers. This set is created by starting with 1 and 1 so the third number becomes 2 and then the cycle continues till infinity. The fibonacci series starts from 1,1,2,3,5,8,13,21……..and goes till infinity.

#### Applications of Number Theory

Number theory enables us to determine whether an integer ‘m’ is divisible by another integer ‘n’ through different divisibility tests. The principles of number theory are not restricted to mathematics, these concepts are used for numerous tasks like:

• Security System in banking securities
• E-commerce websites
• Coding theory
• Barcodes
• Making of modular designs
• Memory management system
• Authentication system

#### Sample Questions

1. If a-b=11 and ab=15 such that a≠0, b≠0, then \frac{1}{a}-\frac{1}{b} = ?
1. \frac{11}{15}

2. \frac{4}{15}

3. \frac{15}{11}

4. -\frac{15}{11}

5. -\frac{11}{15}

Explanation:

Given that,

a-b=11 and ab=15

\frac{1}{a}-\frac{1}{b} = \frac{b-a}{ab}

We know that,

a-b=11

So,

b-a=-11

\frac{1}{a}-\frac{1}{b} = \frac{b-a}{ab} = -\frac{11}{15}

1. Of the following statements, which of the fractions must be terminating decimal?
1. \frac{1}{2^{100}}

2. \frac{3}{300}

3. \frac{5}{2^{20}}
1. I only
2. II only
3. I and II only
4. II and III only
5. I, II and III

Explanation:

Terminating fraction:

A fraction in its simplest form is called a terminating fraction, if it has only 2’s or 5’s (both 10’s) in the denominator.

Example: 1/8 or 1/25 is a terminating fraction.

So, in this question,

\frac{1}{2^{100}} ? Terminating. It has only 2’s in the denominator. So eliminate B.

\frac{3}{300} ? \frac{1}{100} Terminating. It has only 2’s and 5’s in the denominator. So eliminate A and D.

\frac{5}{2^{20}} ? Terminating. It has only 2’s in the denominator. So eliminate C.