Number Theory

Prime and composite numbers, Even and odd numbers

Basic concepts of Mathematics

• Binary Number System

A binary number system has only two digits, 0 and 1. Each of these digits is known as bit.

The normal number system that we use in our daily life has the decimal number system with base 10 while the binary number system has base 2.

For example, 4 can be written in binary number system as 100. Mathematically, it can be represented as (4)₁₀ = (100)₂.

• Conversion of a Binary Number to a Decimal Number

To convert a binary number to a decimal number, we expand the binary number in the powers of 2 according to the place values of digits. On simplifying the expansion, we get the decimal number.

For example, let us convert (11010. 011)₂ into decimal number.

$$\begin{gathered} {\left(11010.011\right)}_2 \\ =1\times 2^4+1\times 2^3+0\times 2^2+1\times 2^1+0\times 2^0+0\times 2^{-1}+1\times 2^{-2}+1\times 2^{-3} \\ =16 + 8 + 0 + 2 + 0 + 0 + 0.25 + 0.125 \\ =26.375 \end{gathered}$$

• Conversion of a Decimal Number to a Binary Number

To convert a decimal number to a binary number, we divide the decimal number continuously by 2 until we get 0 or 1 as the last remainder and keep writing the remainders of each step separately. The reverse order of the remainders is the required binary number.

For example, let us convert (35)₁₀ into binary number.

2	35	Remainder
2	17	1
2	8	1
2	4	0
2	2	0
2	1	0
	0	1

$\mathrm{So}, {\left(35\right)}_{10} = {\left(100011\right)}_2$

 

• Concept Related to Unit digits of Numbers

Unit digits of exponential numbers follow a particular sequence. After a certain number of digits, the sequence gets repeated.

For example,

For powers of 2, we have

2¹ = 2 = unit digit is 2

2² = 4 = unit digit is 4

2³ = 8 = unit digit is 8

2⁴ = 16 = unit digit is 6

2⁵ = 32 = unit digit is 2

...

It can be observed that powers of 2 follow the order 2,4,8,6.

Similarly, the orders of powers of different digits are given as follows:

Digit	Unit digit according to powers
		4n	4n + 3	4n + 2	4n + 1
2		6	8	4	2
3		1	7	9	3
4		6	4	6	4
5		5	5	5	5
6		6	6	6	6
7		1	3	9	7
8		6	2	4	8
9		1	9	1	9

Using these results, we can find the unit digits of larger numbers.

For example,

Unit digit of (456)²⁴⁵ = Unit digit of (456)^{64×4 + 1} = 6.

• Properties of Logarithms

Solved Examples

 

Example 1: Convert (101101.101)₂ to decimal number.

Solution :

  $$\begin{gathered} {\left(101101.101\right)}_2 \\ =1\times 2^5+0\times 2^4+1\times 2^3+1\times 2^2+0\times 2^1+1\times 2^0+1\times 2^{-1}+0\times 2^{-2}+1\times 2^{-3} \\ =32 + 0 + 8 + 4 + 0 + 1 + 0.5 + 0 + 0.125 \\ =45.625 \end{gathered}$$

Example 2: What is the least positive remainder when 7¹³³ is divided by 5?

Solution:

We know that the unit place of powers of 7 repeats after every fourth power.

Now 7¹³³ = (7)^{4 ×33 + 1}

Unit digit of (7)^{4n+ 1} = 7

∴ Unit digit of (7)^{4 ×33 + 1} = 7

On dividing by 5, 7 gives the remainder 2.

Hence, required remainder when 7¹³³ is divided by 5 is 2.

The numeral system we usually use has ten digits, from 0 to 9; for this reason, it is known as the base 10 system or the decimal number system. In this system of numeration, we create numbers using the ten digits. For example, 82 and 1024 are decimal numbers. These numbers can also be written as 82₍₁₀₎ and 1024₍₁₀₎.

Another system of numeration is the base 2 system or the binary number system. In this system, we form numbers using the two digits “0” and “1”. For example, 0₍₂₎, 1₍₂₎, 10₍₂₎, 100₍₂₎ and 101₍₂₎ are binary numbers. These are read as “zero to the base 2”, “one to the base 2”, “one zero to the base 2”, “one zero zero to the base 2” and “one zero one to the base 2”, respectively.

The following table lists a few numbers written as per the base 10 and base 2 systems.

Things	Unit of Things	Decimal Representation	Binary Representation
	zero	0	0
*	one	1	1
**	two	2	10
***	three	3	11
****	four	4	100
*…