# Playing With Numbers

### Prime and Composite Numbers

• An exact divisor of a number is called its factor.
• The number 1 is a factor of every number.
• Every number is a factor of itself.
• The factors of a number are either less than or equal to the number itself.
• All numbers have a finite number of factors.
• The product of two numbers is called a multiple of each of the two numbers being multiplied.
• A number is a multiple of all its factors.
• Every number is a multiple of 1 and itself.
• There are infinite multiples of a number.
• Numbers that have only two factors in the form of 1 and the number itself are called prime numbers.
• Numbers that have more than two factors are called composite numbers.
• The number 1 is neither a prime number nor a composite number.
• All number having 0, 2, 4, 6 or 8 in the unit‘s or one‘s place are multiples of 2 and are called even numbers.
• All numbers having 1, 3, 5, 7 or 9 in the unit‘s place are called odd numbers.
• The number 2 is the smallest prime number and also the only prime number that is even.
• All prime numbers, except 2, are odd numbers.
• The sum of any two prime numbers, except 2, is an even number.

### Divisibility of Numbers

• A number that has 0, 2, 4, 6 or 8 in its place is divisible by 2.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• A number that has 0 or 5 in its ones place is divisible by 5.
• A number is divisible by 6 if that number is divisible by both 2 and 3.
• A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
• A number is divisible by 9 if the sum of its digit is divisible by 9.
• A number that has 0 in its ones place is divisible by 10.
• If the difference between the sum of digit at odd and even places in a given number is either 0 or a multiple of 11, then the given number is divisible by 11.
• If a number is divisible by another number, then it is also divisible by all factors of the other number.
• If two numbers are divisible by another number, then their sum and difference is also divisible by the other number.
• If a number is divisible by two co-prime numbers, then it is also divisible by the product of the two co-prime numbers.
• If the only common factor of two numbers is 1, then the two numbers are called co-prime numbers.

### Prime Factorisation, HCF and LCM

• Writing a number as a product of its prime factors is called prime factorization of the number.
• The greatest common factor of given numbers is called their highest common factor (HCF).
• The HCF of given numbers is equal to the product of their common prime factors in prime factorization.
• The lowest common multiple of given numbers is called their Least Common Multiple (LCM).
• The LCM of given number is equal to the product of their prime factors counted as the maximum number of times they occur in their prime factorization.

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