# Tutorial on Integers

The numbers -3,-2,-1,0,1,2,3 are all integers. -1,-2, and -3 are negative integers. Similarly +1,+2,+3 are positive integers, however we write them as 1,2 ,3 leaving out the + sign. 0 is also an integer.

Multiples
If we multiply 5 by 3 the answer will be 15. We may write this as:

5 x 3 = 15

We say that 15 is a multiple of both 5 and 3;

Factors
12 can be divided by any of the following numbers:

1,2,3,4,6 and 12 itself

All these numbers are called factors of 12

Prime Numbers
A prime number has only two factors, these are 1 and the prime number itself. Examples of prime numbers are:

3 its factors are 1,3
5 its factors are 1,5
7 its factors are 1,7
and so on

Any integer greater than 1 is either a prime number or can be written as a product of prime numbers. For example:

2: a prime number
3: a prime number
4= 2 x 2: 2 is a prime number
5: a prime number
6= 2 x 3: 2 and 3 are prime numbers
7: a prime number
8= 2 x 2 x 2: 2 is a prime number
9= 3 x 3: 3 is a prime number
10= 2 x 5: 2 and 5 are prime numbers
and so on

Common Multiples
8 is a common multiple of 2 and 4
10 is a common multiple of 2 and 5
12 is a common multiple of 2,3,4 and 6

Is 18 a common multiple of 6 and 7 ? The answer is no, as though
6 x 3 = 18, 7 cannot be multiplied by any number to give 18, so 18 is not a multiple of 7, though it is of 6.

Least Common Multiple (LCM)
As the name suggests the LCM of two numbers is the smallest number which is a multiple of both numbers.

Method for Finding LCM
Let us take two numbers 6 and 9

• Step 1
Write the number as a product of prime numbers:
so 6 = 2 x 3 and 9 = 3 x 3

• Step 2
Look for common factors and delete them in one of the products. In our example 3 is a common factor, so delete on of the 3s. This leaves us with 2,3 and 3

• Step3
Now we multiply these remaining factors, thus in our example we get:
2 x 3 x 3 = 18
which gives us 18 as the LCM of 6 and 9