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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.
Note:
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.
Note:
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.
Note:
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.
Note:
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.
Note:
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.

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