Number System: Classification and Divisibility Rules

By | January 7, 2016

Classification of Numbers:

Natural numbers:

All the counting numbers are classified as Natural numbers.

N = {1,2,3,….}

Whole numbers:

All natural numbers and the number zero (0) are classified under Whole numbers.

W = {0,1,2,3,4,….}

Integers:

Integers are all of the whole numbers and the negative natural numbers are classified under Integers.

I = {…,-3,-2,-1,0,1,2,3,….}

Rational numbers:

A rational number is a number that can be written as a ratio of form p/q where p and q are integers, q ≠ 0.

Any terminating or recurring decimal is a rational number.

E.g: 5/3, 1.5, 4 etc.

Irrationals:

An irrational number that cannot be written in the form p/q. Any non-terminating and non-recurring decimals is an irrational number.

E.g:

π     = 3.141592…

Square root of 2 = 1.414213…

Real numbers:

All rational and Irrational numbers together are classified under Real numbers.

Decimal numbers:

The zero and the counting numbers (1,2,3,…) make set of whole numbers. But not every number is a whole number. Our decimal system let us write numbers of all types and sizes, using a clever symbol called point.

As you move right from decimal point, each place value is divided by 10.

Even numbers:

Numbers that are divisible by 2 are even numbers.

E.g. 2, 4, 10, 120, -14 etc.

Odd numbers:

Numbers that are not divisible by 2 are odd numbers.

E.g: 1, 3, 7, 15, -9 etc.

Prime numbers:

Numbers that do not any have any factors apart from one and the number itself.

E.g: 2,3,5,7,11,13,17,19,23 etc.

Composite numbers:

Numbers that have more than 2 factors are composite numbers.

Coprimes or relative primes:

Numbers that do not have any common  actor to them except 1 are coprimes or relative primes.

E.g: 4 and 9 are coprimes.

Properties of even and odd numbers

  • The sum of any numbers of even numbers is always even.
  • The product of any number of even number is always even.
  • The sum of odd number of odd numbers is always odd.
  • The sum of even number of odd numbers is always even.
  • The product of any number of odd number is always odd.

Divisibility Rules:

Divisibility by 3:

A number is divisible by 3 if he sum of digits is multiple of 3.

E.g: 27, 126, 3729 etc.

Divisibility by 4:

A number is divisible by 4 if the number formed with its last two digits is divisible by 4.

E.g: 112, 3428, 36836 etc.

Divisibility by 5:

A number is divisible by 5 if its last digit is 5 or zero.

Divisibility by 6:

A number is divisible by 6 if it is divisible by both 2 and 3.

Divisibility by 7:

A number is divisible by 7 if the difference between the number of tens in the number and twice the unit’s digit is divisible by 7.

Divisibility by 8:

A number is divisible by 8, if the number formed with the last 3 digits of the number is divisible by  8.

Divisibility by 9:

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility by 10:

A number is divisible by 10 if its last digit is zero.

Divisibility by 11:

A number is divisible by 11 if the difference between the sum of digits in odd places in the number and the sum of the digits in the even places in the number should be equal to zero or multiple of 11.

Prime numbers to 100

Here is the list of prime numbers upto 100. You can see that none of these numbers can be factored any further.

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97

A composite number is any number that has more than two factors.

You can write any composite number as a product of prime factors. this is called prime factorization.

Factors and multiples:

In the problem 3*4 = 12,

3 and 4 are factors and 12 is the product or multiple.

Factoring like taking a number apart. It means to express a number as the product of its factors.

Factors are either composite numbers or composite numbers (except that 0 and 1 are neither prime nor composite).

The number 12 is a multiple of 3, because it can be divided evenly by 3.

3*4 = 12.

3 and 4 are both factors of 12.
12 is a multiple of both 3 and 4.

To find the factors – it is like multiplying in reverse. here are lists of all the factors of 16,20 and 45.

16 –> 1, 2, 4, 8, 16
20 –> 1, 2, 4, 5, 10, 20
45 –> 1, 3, 5, 9, 15, 45.

Greatest Common Factor (GCF):

The greatest common factor, or GCF, is the greatest factor that divides two numbers. To find the GCF of two numbers:

List the prime factors of each number.

Multiply those factors both numbers have in common. If there are no common prime factors, the GCF is 1.

Solved Example: Lets find GCF of 30 and 45.

First we find the prime factors of each number, using prime factorization.

30 = 2*3*5
45 = 3*3*

Next, identify those prime factors that both numbers have in common and multiply them. Here, both 3 and 5 are common factors. the GCF is 3 times 5, or 15.
GCF = 3*5 = 15

Least Common Multiple (LCM):

A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 are 0, 12, 24,…
The least common multiple  (LCM) of two numbers is the smallest number ( not zero) that is a multiple of both.

Solved example: Let’s find the LCM of 30 and 45.

One way to find the least common multiple of two numbers id to first list the prime factors of each number.

30 = 2*3*5
45 = 3*3*5

Then multiply each factor the greatest number of times it occurs in either number, If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs.

2: one occurence
3: two occurences
5: one occurence

LCM = 2*3*3*5 = 90

After you have calculated a least common multiple, always check to be sure your answer can be divided evenly by two numbers.