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.
(ii) Odd number : Numbers which are not multiples of 2 are called odd numbers.
(iii) Prime numbers : Each of numbers which have exactly two factors namely, 1 and number itself, are 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.
(iv) Composite numbers : numbers having more than two factors are known as 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.
(v) Twin prime : Two consecutive odd prime numbers are known as twin prime, e.g.,
(i) 3, 5 (ii) 5, 7 (iii) 11, 13 etc.
(vi) Prime triplet : A set of three consecutive prime numbers, differing by 2, is called a prime triplet. only prime triplet is (3, 5, 7).
(vii) Co-primes : Two numbers are said to be co-prime if they do not have a common factor other than 1. e.g.
(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.
(viii) Perfect numbers : If sum of all factors of a number is two times number, then number is called 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.
Note : Some facts about factors and Multiples
• Every number has two factors at least which are 1 and number itself.
• 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.
Chapter Notes and Summary