Number System

The Number System is defined as the branch of mathematics that deals with the study of different types of numbers, their classification, properties, and the rules governing operations on them.

Type of Number System:

1. Natural Numbers: The set of positive integers starting from 1 and going on infinitely.

Example: N = {1, 2, 3, 4, 5...}

• Even Numbers: An even number is any number that gives a remainder of 0 when divided by 2.

Example: 2,4,6,8...

General Form: Even number 2 x n, where n is an integer.

• Odd Numbers: An odd number is a number that leaves a remainder of 1 when divided by 2.

Example: 1,3,5,7,9,11...

General Form: Odd number 2 x n + 1, where n is an integer.

• Prime Numbers: Prime numbers are natural numbers greater than 1 that have exactly two distinct factors- 1 and the number itself.

Example: 2,3,5,7,11,13,17...

2 is the only even prime number.

• Co-primes: Co-primes are two numbers that have no common factors other than 1. In other words, they do not share any divisor except 1.

Example: 14 and 25

• Composite Numbers: A composite number can be divided exactly by 1, itself, and at least one other number.

Example: 4, 6, 8, 9, 10...

1 is not a composite number

2. Whole Numbers: The set of numbers that includes all natural numbers and the number zero is called whole numbers.

Example: 0,1,2,3,4...

They do not include negative numbers or fractions.

0 is the smallest whole number.

3. Integers: Integers are whole numbers that can be positive, negative, or zero, but not fractions or decimals.

Example: -4, -3, -2, -1, 0, 1, 2, 3, 4...

4. Rational Numbers: Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero.

A rational number is any number of the form p/q, where p and q are integers and q ≠ 0.

Example: 1/2, 3/4, 4, -6 etc.

5. Irrational Numbers: Irrational numbers are numbers that cannot be expressed as a fraction (p/q) where p and q are integers and q ≠ 0.

An irrational number is a number with a non-terminating, non-repeating decimal expansion.

Example: √2= 1.414..., √5= 2.236...

6. Real Numbers: A real number is any number that is either rational or irrational.

Some Important Formula:

1. a²- b² = ( a + b ) ( a - b )

2. ( a + b )² = a² + b² + 2ab

3. ( a - b ) = a² + b² - 2ab

4. ( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ac

5. ( a + b )³ = a³ + b³ + 3ab ( a + b )

6. ( a - b )³ = a³ - b³ - 3ab ( a - b)

7. a³ + b³ = ( a + b ) ( a² + b² - ab )

8. a³ - b³ = ( a - b ) ( a² + b² + ab )

9. a³ + b³ + c³ - 3abc = ( a + b + c ) ( a² + b² + c² - ab - bc - ac )

Divisibility Rules

Divisibility by 2:
A number is divisible by 2 if its last digit is even.
Example: 140, 158

Divisibility by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example: 123 → 1 + 2 + 3 = 6 → 6 is divisible by 3, hence 123 is divisible by 3.

Divisibility by 4:
A number is divisible by 4 if the last two digits are divisible by 4.
Example: The last two digits of 1224 are 24. Since 24 is divisible by 4, 1224 is divisible by 4.

Divisibility by 5:
A number is divisible by 5 if its last digit is 0 or 5.
Example: The last digit of 1520 is 0, so 1520 is divisible by 5.

Divisibility by 6:
A number is divisible by 6 if it is divisible by both 2 and 3.
Example: 786 ends in an even number (divisible by 2), and the sum of its digits (7 + 8 + 6 = 21) is divisible by 3. Hence, 786 is divisible by 6.

Divisibility by 7:
Double the unit digit and subtract it from the rest of the number. If the result is divisible by 7, then the number is divisible by 7.
Example: The unit digit of 875 is 5. Double it: 5 × 2 = 10. The remaining number is 87. Now, 87 − 10 = 77 → 77 is divisible by 7. Hence, 875 is divisible by 7.

Divisibility by 8:
A number is divisible by 8 if the last three digits are divisible by 8.
Example: 4512 → last three digits are 512 → 512 ÷ 8 = 64 → divisible by 8.

Divisibility by 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example: 585 → 5 + 8 + 5 = 18 → divisible by 9.

Divisibility by 10:
A number is divisible by 10 if its last digit is 0.
Example: The last digit of 430 is 0 → divisible by 10.

Divisibility by 11:
A number is divisible by 11 if the difference between the sum of digits at odd positions and even positions is 0 or a multiple of 11.
Example: 1661 → (1 + 6) = 7 (odd positions), (6 + 1) = 7 (even positions) → 7 − 7 = 0 → divisible by 11.

Divisibility by 12:
A number is divisible by 12 if it is divisible by both 3 and 4.
Example: 132 → 1 + 3 + 2 = 6 (divisible by 3), last two digits 32 (divisible by 4) → 132 is divisible by 12.

Divisibility by 13:
Multiply the unit digit by 4, then add it to the rest of the number. If the result is divisible by 13, then the number is divisible by 13.
Example: 351 → unit digit is 1 → 1 × 4 = 4, remaining number is 35 → 35 + 4 = 39 → divisible by 13.

Divisibility by 14:
A number is divisible by 14 if it is divisible by both 2 and 7.
Example: 126
→ Ends in 6 → divisible by 2
→ 126 ÷ 7 = 18 → divisible by 7
→ 126 is divisible by 14

Divisibility by 15:
A number is divisible by 15 if it is divisible by both 3 and 5.
Example: 135
→ 1 + 3 + 5 = 9 → divisible by 3
→ Ends in 5 → divisible by 5
→ 135 is divisible by 15

Divisibility by 16:
A number is divisible by 16 if the last 4 digits (or the entire number if ≤ 4 digits) form a number divisible by 16.
Example: 1600
→ Last 4 digits = 1600 → 1600 ÷ 16 = 100
→ 1600 is divisible by 16

Divisibility by 17:
Multiply the last digit by 5 and subtract from the remaining digits. If the result is divisible by 17, the number is divisible. (Example not provided in original.)

Divisibility by 18:
A number is divisible by 18 if it is divisible by both 2 and 9.

Divisibility by 19:
Multiply the last digit by 2 and subtract from the rest. If the result is divisible by 19, the number is divisible.

Divisibility by 20:
A number is divisible by 20 if it ends in 00, 20, 40, 60, or 80 (i.e., divisible by both 4 and 5).

Important Concept

1. HCF (Highest Common Factor)

Definition:
The HCF of two or more numbers is the greatest number that divides each of them exactly.

Methods to Find HCF:

a. Prime Factorization Method:

Find the prime factorization of each number.
Take the common prime factors with the lowest powers.
Multiply them.
Example:
12 = 2² × 3
18 = 2 × 3²
HCF = 2¹ × 3¹ = 6

b. Division Method (Euclid’s Algorithm):

Divide the larger number by the smaller.
Divide the divisor by the remainder.
Repeat until the remainder is 0.
The last divisor is the HCF.

2. LCM (Least Common Multiple)

Definition:
The LCM of two or more numbers is the smallest number that is divisible by all of them.

Methods to Find LCM:

a. Prime Factorization Method:

Find the prime factorization of each number.
Take all prime factors with the highest powers.
Multiply them.
Example:
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 36

b. Common Multiple Method (for small numbers):

List some multiples of each number.
Find the first common multiple.

3. Important Relationship Between HCF and LCM

For two numbers:
HCF × LCM = Product of the two numbers

Example:
Given Numbers: 12 and 18
HCF = 6, LCM = 36
6 × 36 = 12 × 18 = 216

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