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. 11010.0112=1×24+1×23+0×22+1×21+0×20+0×2-1+1×2-2+1×2-3=16 + 8 + 0 + 2 + 0 + 0 + 0.25 + 0.125=26.375

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 1So, 3510 = 1000112

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 :

101101.1012=1×25+0×24+1×23+1×22+0×21+1×20+1×2-1+0×2-2+1×2-3=32 + 0 + 8 + 4 + 0 + 1 + 0.5 + 0 + 0.125=45.625

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.

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. 11010.0112=1×24+1×23+0×22+1×21+0×20+0×2-1+1×2-2+1×2-3=16 + 8 + 0 + 2 + 0 + 0 + 0.25 + 0.125=26.375

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 1So, 3510 = 1000112

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 :

101101.1012=1×25+0×24+1×23+1×22+0×21+1×20+1×2-1+0×2-2+1×2-3=32 + 0 + 8 + 4 + 0 + 1 + 0.5 + 0 + 0.125=45.625

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

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