Playing with Numbers

Prime and Composite Numbers

An exact divisor of a number is called its factor.

An exact divisor of a number is called its factor.

Ex: 1, 2, 3 and 6 are factors of number 6.

F The number 1 is a factor of every number.

F Every number is a factor of itself.

F The factors of a number are either less than or equal to the number itself.

F All numbers have a finite number of factors.

F The product of two numbers is called a multiple of each of the two numbers being multiplied.

F A number is a multiple of all its factors. Every number is a multiple of 1 and of itself.

F There are infinite multiples of a number.

F If the sum of the factors of a number is two times the number, then the number is called a perfect number. 

F Numbers that have only two factors in the form of 1 and the number itself are called prime numbers.

F Numbers that have more than two factors are called composite numbers.

F The number 1 is neither a prime number nor a composite number.

F All numbers with 0, 2, 4, 6 or 8 in the unit's or one's place are multiples of 2, and are called even numbers.

F All numbers with 1, 3, 5, 7 or 9 in the unit's or one's place are called odd numbers.

F The number 2 is the smallest prime number, and also the only prime number that is even.

F All prime numbers, except 2, are odd numbers.

F The sum of any two prime numbers, except with 2, is an even number.

Divisibility of Numbers

Factor: 

An exact divisor of a number is called its factor.

Multiple: 

The product of two numbers is called a multiple of each of the two numbers being multiplied.

Prime numbers: 

Numbers with two factors, 1 and itself, are called prime numbers.

Tests of divisibility:

There are certain tests of divisibility that can help us to decide whether a given number is divisible by another number.

Ø Divisibility of numbers by 2
A number that has 0, 2, 4, 6 or 8 in its ones place is divisible by 2.

Ø Divisibility of numbers by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.

Ø Divisibility of numbers by 4
A number is divisible by 4 if the number formed by its last two digits (i.e. ones and tens) is divisible by 4.

Ø Divisibility of numbers by 5
A number that has either 0 or 5 in its ones place is divisible by 5.

Ø Divisibility of numbers by 6
A number is divisible by 6 if that number is divisible by both 2 and 3.

Ø Divisibility of numbers by 8
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Ø Divisibility of numbers by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

Ø Divisibility of numbers by 10
A number that has 0 in its ones place is divisible by 10.

Ø Divisibility of numbers by 11
If the difference between the sum of the digits at the odd and even places in a given number is either 0 or a multiple of 11, then the given number is divisible by 11.

Co-prime numbers:

If the only common factor of two numbers is 1, then the two numbers are called co-prime numbers.

General rules of divisibility for all numbers:

If a number is divisible by another number, then it is also divisible by all the 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.

Prime Factorization, HCF and LCM

Ø Writing a number as a product of its prime factors is called the prime factorisation of the number.

Eg:

(i) 18 = 2 x 3 x 3 
(ii) 40 = 2 x 2 x 2 x 5

Ø The greatest of the common factors of the given numbers is called their highest common factor (HCF). It is also known as the greatest common divisor.

Eg:

Prime factorisation of 16 = 2 x 2 x 2 x 2
Prime factorisation of 40 = 2 x 2 x 2 x 5
HCF of 16 and 40 = 2 x 2 x 2 = 8

Ø The smallest common multiple of the given numbers is called their Least Common Multiple (LCM).

Eg:

The LCM of given numbers using their prime factorisation:
Prime factorisation of 4 = 2 x 2
Prime factorisation of 6 = 2 x 3
LCM of 4 and 6 = 2 x 2 x 3 =12

To find the LCM of the given numbers using the division method:

Ø Write the given numbers in a row.

Ø Divide the numbers by the smallest prime number that divides one or more of the given numbers.

Ø Write the number that is not divisible, in the second row.

Ø Write the new dividends in the second row.

Ø Divide the new dividends by another smallest prime number.

Ø Continue dividing till the dividends are all prime numbers or 1.

Ø Stop the process when all the new dividends are prime numbers or 1.