**Chapter Notes and Summary**

• **Factor : **A factor of a number is an exact divisor of that number.•

**Multiple :**A number is said to be a multiple of any of its factors.

**•**

**Various types of numbers :**

(i) Even number :All multiples of 2 are called even number.

(i) Even number :

**Numbers which are not multiples of 2 are called odd numbers.**

(ii) Odd number :

(ii) Odd number :

**Each of numbers which have exactly two factors namely, 1 and number itself, are**

(iii) Prime numbers :

(iii) Prime numbers :

**called prime numbers, e.g., 2, 3, 5, 7, 11, ……….etc. 2 is smallest prime number which is even. Every prime**

**number except 2 is odd.**

**numbers having more than two factors are known as composite numbers.**

(iv) Composite numbers :

(iv) Composite numbers :

**e.g., 4, 6, 8, 9, 10, 12, 14,……. etc. and 1 is neither prime nor composite because it has a single factor.**

**Two consecutive odd prime numbers are known as twin prime, e.g.,**

(v) Twin prime :

(v) Twin prime :

**(i) 3, 5 (ii) 5, 7 (iii) 11, 13 etc.**

**A set of three consecutive prime numbers, differing by 2, is called a prime triplet. only**

(vi) Prime triplet :

(vi) Prime triplet :

**prime triplet is (3, 5, 7).**

**Two numbers are said to be co-prime if they do not have a common factor other than 1. e.g.**

(vii) Co-primes :

(vii) Co-primes :

**(2, 3) (3, 4), (4, 5) etc. Any two prime numbers are always co-prime but converse is not always true i.e. both**

**co-prime numbers need not be prime.**

**If sum of all factors of a number is two times number, then number is called**

(viii) Perfect numbers :

(viii) Perfect numbers :

**a perfect number e.g., 6.**

**•**

**Divisibility Test :**

(i) A number is divisible by 2, if its unit digit are 0, 2, 4, 6 or 8.

**(ii) A number is divisible by 3, if sum of its digits is divisible by 3.**

**(iii) A number is divisible by 4, if number formed by last two digits is divisible by 4.**

**(iv) A number is divisible by 5, if its unit digit is 0 or 5.**

**(v) A number is divisible by 6, if number is divisible by 2 and 3.**

**(vi) A number is divisible by 8, if number formed by last three digits is divisible by 8.**

**(vii) A number is divisible by 9, if sum of its digits is divisible by 9.**

**(viii) A number is divisible by 11, if difference of sum of its digits in odd places and sum of its digits in**

**even places is either 0 or a multiple of 11.**

**•**

**Highest Common Factor :**(HCF) of two or more number is highest of their common factors. If is also known

**as Greatest common divisor (GCD).**

**• Least Common Multiple (LCM) of two or more numbers is lowest of their common multiples.**

**• HCF × LCM = Product of numbers. This property is valid for two numbers only.**

**• Every number has two factors at least which are 1 and number itself.**

Note : Some facts about factors and Multiples

Note : Some facts about factors and Multiples

**• Every factor of a number is an exact divisor of that number.**

**• Every factor is less than or equal to given number.**

**• There are finite number of factors of a given number.**

**• Every multiple of a number is greater than or equal to that number.**

**• number of multiples of a given number is infinite.**

**• Every number is a multiple of itself.**