Even Numbers: All multiples of 2 are called even numbers. For eg. 2 ,4 , 6 , ….are even numbers
Odd Numbers: Numbers which are not multiples of 2 are odd numbers.For eg. 1, 3 ,5 ,7,….
Prime Numbers: Each number which has two factors 1 and itself is called a prime number. For eg. 3 , 5 ,7 ……….etc.
Composite numbers: Numbers which have more than two factors are called composite numbers. For eg.
4 ,6 ,8,9 ….etc.
1. 1 is neither prime nor composite
2. 2 is the smallest prime number and is the only even prime number.
Twin Prime Numbers: Two prime numbers whose difference is 2 are called twin prime numbers.
Eg, (3 ,5), (11,13), ……..
Prime triplet: A set of three consecutive prime numbers ,differing by 2 is called a prime triplet. For eg.(3,5,7) is prime triplet.
Perfect Numbers: If the sum of all the factors of a number is two times the number then the number is called the perfect number.
For eg. 6 is a perfect number. Factors of 6 are 1,2,3,6 . Sum of factors of 6 is 1+2+3+6=12 = 2 X 6.
Co – primes: Two numbers are said to be co-prime if they do not have a common factor other than 1.
For eg. 2,3 are co-primes . 3 ,4 are co-primes etc
Co-prime numbers are represented as (a,b) =1 meaning HCF of co-prime numbers is 1.
For eg. (2,3)=1.
1. H.C.F. or G.C.D. of two co-prime numbers is always 1.
2. Two prime numbers are always co-primes.
Factor: A factor of a number is the exact divisor of that number. For eg. Factors of 6 are 2 and 3.
Multiple: A number is said to be a multiple of each of its factors. For eg. 5 X 6 = 30 where 5 and 6 are factors of 30 and 30 is a multiple of both 5 and 6.
1. 1 is the factor of every number.
2. Every number is a factor of itself.
3. Every factor of a number is an exact divisor of that number.
4. Every factor is less than or equal to the given number.
5. Number of factors of a given number are finite.
6. Every multiple of a given number is greater than or equal to that number.
7. The number of multiples of a given number are infinite.
8. Every number is multiple of itself.
Tests of divisibility
Divisibility by 10: If a number has 0 in ones place it is divisible by 10. Eg 10 , 100, 1000 etc are divisible by 10 as all of them have 0 in units place.
Divisibility by 5: If a number has 0 or 5 in its ones place it is divisible by 5. For eg. 5, 10 ,15 ,20 ,25 etc are divisible by 5.
Divisibility by 2: A number is divisible by 2 if it has any of the digits 0 ,2 ,4 ,6, 8 in its ones place.
For eg . 12 , 24 , 28 etc are divisible by 2.
Divisible by 3: A number is divisible by 3 if the sum of the digits of the number is a multiple of 3.
For eg. 7536 is divisible by 3 because 7+5+3+6 = 21 which is multiple if 3.
Divisibility by 4: A number is divisible by 4 is the number formed by the last two digits of the number is divisible by 4.
For eg . 9984 is divisible by 4 because the number formed by the last two digits is 84 which is divisible by 4.
Divisibility by 6: A number is divisible by 6 if it is divisible by 2 and 3 both. For eg , 59634 is divisible by 2 as the last digit of the number is even and it is divisible by 3 as 5+9+6+3+4 = 27 which is multiple of 3 .Thus , 59634 is divisible by 6 also.
Divisibility by 8: A number with 4 or more digits is divisible by 8, if the number formed by the last three digits is divisible by 8. For eg . 75816 is divisible by 8 because 816 is divisible by 8.
Divisibility by 9: If the sum of the digits of a number is divisible by 9 then the number itself is divisible by 9.
For eg. 8406 is divisible by 9 as 8+4+0+6 = 18 which is divisible by 9.
Divisibility by 11: Find the difference between the sum of the digits at odd places (from the right) and the sum of the digits at the even places( from the right ) of the number. If the difference is either 0 or multiple of 11 , then the number is divisible bu 11.
For eg. 1331 is divisible by 11 as Sum of digits in odd places = 1+3 = 4 and Sum of digits is even places= 1+3=4. Difference between the two sums= 4-4=0.
1. If a number is divisible by another number then it is divisible by each of the factors of that number.
2. If a number is divisible by two co-primes then it is divisible by their product also.
3. If two given numbers are divisible by a number, then their difference is also divisible by that number.
Highest Common Factor (HCF): The highest common factor (H.C.F.) of two or more given numbers is the highest (or greatest) of their common factors.
Lowest Common multiple (LCM): The lowest common multiple of two or more given numbers is lowest (or smallest or least) of their common multiples.
1. HCF of given numbers is never greater than any of the given numbers.
2. The HCF of two co-prime numbers is 1.
3. The LCM of given numbers is never less than any of the given numbers.
4. The LCM of two co-prime numbers is equal to their product.
5. The HCF of given numbers is always a factor of their LCM.
6. If two numbers are given the the product of the two numbers is equal to the product of their LCM and HCF.
number1 X number1 = LCM X HCF
7. If x and y are two numbers such that x is a factor of y then the HCF of the two numbers = x and LCM of the two numbers = y.
Division of a number by 0: Division of a number by 0 is not defined in mathematics. For eg .
0/5 = 0 but, 5/0 = not defined.