Special types of numbers

​​In a nutshell

All the numbers that you use can be categorised into different groups.

Integers

Integers are whole numbers - both positive and negative.  Zero is also considered an integer.

Odd and even numbers

Odd numbers are integers that end in $$1,3,5,7,9$$​. They all leave a remainder of $$1$$​ when divided by $$2$$​.

Even numbers are integers that end in $$0,2,4,6,8$$​. They are all divisible by $$2$$​.

Rational and irrational numbers

Rational numbers are numbers that can be written as fractions where the numerator and denominator are both integers.  Examples of rational numbers include:

​$$-\frac{1}{6},\frac{500}{49},12\,(=\frac{12}{1})$$​​

Irrational numbers are numbers that can't ​be written as fractions where the numerator and denominator are both integers. Examples of irrational numbers include $$\pi$$ and any surd.

Square and cube numbers

Square numbers are integers that can be written as the square of another integer.

The first ten square numbers are:

Cube numbers are integers that can be written as the cube of another integer.

The first five cube numbers are:

Prime numbers

A prime number is a positive integer that is only divisible by $$1$$​ and itself.

Key facts about prime numbers

• ​$$1$$​ is not a prime number - a prime number has to have two different factors (numbers it is divisible by).
• ​$$2$$​ is the only even prime - all the other even numbers are divisible by $$2$$​ and hence not prime.
• The single digit prime numbers are $$2,3,5,7$$​.
  
• All prime numbers greater than $$10$$​ end in $$1,3,7,9$$.
  
• Just because a number ends in $$1,3,7,9$$, doesn't necessarily mean it is prime. For example, $$33$$ isn't prime as it is divisible by $$3$$​.
• There are an infinite number of primes.
  

Finding two digit prime numbers

It is very difficult to find large prime numbers, as larger numbers have a higher chance of having additional factors. However, for two digit numbers, there is a procedure.

procedure

​​​​

Note: This procedure only works for two digit numbers.

Example

Verify that $$71$$​ is prime.

It ends in $$1$$​. So, check if it is divisible by $$3$$ and $$7$$:

​$$71\div3=23$$​ remainder $$2$$.

$$71\div7=10$$​ remainder $$1$$.

Therefore, ​$$\underline{71}$$ is prime because it ends in $$\underline1$$ and is NOT divisible by $$\underline{3}$$ and $$\underline7$$.