FACTOR
A factor of a given number is every number that divides exactly into that number.
Example:
Write down all factors of 10.
10 = 2 x 5, so numbers 2 and 5 are factors of 10.
Also 10 = 10 x 1, so 10 and 1 are factors of 10.
The factors of 10 are 1, 2, 5, 10.
NOTE: Number 1 and the number itself are always factors of any number.
COMMON FACTORS
When two (or more) numbers have the same factor, that factor is called a common factor.
Example:
Find all the common factors of 12 and 18.
Factors of 12 are 1, 2, 3, 4, 6, 12.
Factors of 18 are 1, 2, 3, 6, 18.
The common factors of 12 and 18 are 1, 2, 3 and 6.
LCM (Least Common Multiple)
The least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then lcm(a, b) is defined to be zero.
Exanple: the L.C.M of 3 and 5 is 15
The simple method of finding the L.C.M of smaller numbers is to write down the multiples of the larger number until one of them is also a multiple of the smaller number.
Example 1:
Find the Lowest Common Multiple of 8 and 12.
Solution: Multiples of 12 are 12, 24...
24 is also a multiple of 8, so the L.C.M of 8 and 12 is 24.
FINDING L.C.M. OF BIG NUMBERS
1. Find all the prime factors of both numbers.
2. Multiply all the prime factors of the larger number by those prime factors of the smaller number that are not already included.
Example:
Find the Lowest Common Multiple (L.C.M.) of 240 and 924.
From the example of finding the H.C.F. we know the prime factors of both numbers.
924 = 2 x 2 x 3 x 7 x 11
240 = 2 x 2 x 2 x 2 x 3 x 5
The L.C.M. is 2 x 2 x 3 x 7 x 11 x 2 x 2 x 5 = 924 x 20 = 18,480
HCF (Highest Common Factor)
Highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder.
It is very easy to find a H.C.F. of small numbers, like 6 and 9 (it is 3) or 8 and 4 (it is 4).
The best way is to keep finding the factors of the smaller number, starting from the largest factor. The first factor of the smaller number that is also a factor of the larger number is a H.C.F.
FINDING THE H.C.F. OF BIG NUMBERS
For larger numbers you can use the following method:
1. Find all prime factors of both numbers.
2. Write both numbers as a multiplication of prime numbers.
3. Find which factors are repeating in both numbers and multiply them to get H.C.F (why?)
Example:
Find the Highest Common Factor (H.C.F.) of 240 and 924.
Solution:
Finding all prime factors of 240:
We will start with the smallest prime number and we will divide 240 into it if we can (the divisibility rules come handy).
We will do the same with the result (or quotient), and we will keep dividing by prime numbers until we have 1 as a quotient. Each time we write the prime factor to the right and the quotient below:
240 | 2 2 is a factor of 240; 240 divided by 2 is 120
120 | 2 2 is a factor of 120; 120 divided by 2 is 60
60 | 2 2 is a factor of 60; 60 divided by 2 is 30
30 | 2 2 is a factor of 30; 30 divided by 2 is 15
15 | 3 3 is a factor of 15; 15 divided by 3 is 5
5 | 5 5 is a factor of 5; 5 divided by 5 is 1
1
240 = 2 x 2 x 2 x 2 x 3 x 5
