1.1 - Factors of Numbers, Greatest Common Factors and Least Common Multiples




Factors of a number

The numbers that we are interested in factoring are the natural numbers 1, 2, 3, … The word factor is used as both a noun and a verb. The factors (noun) of a number are the numbers that divide evenly into the number. For example the factors of the number 12 are the numbers 1, 2, 3, 4, 6 and 12. (Notice that the smallest factor is always 1 and the biggest factor is always the number itself.)

To factor (verb) a number means to express it as a product of smaller numbers. For example we can factor the number 12 like this: 12 = 3 · 4. The numbers 3 and 4 are called the factors. Another way to factor 12 is like this: 12 = 2 · 2 · 3. Now the factors are 2, 2 and 3. Each way of factoring a number is called a factorization.

A number that cannot be factored further is called a prime number. To factor a number completely means to write it as a product of prime numbers. This is also called the prime factorization.

Here are some examples of numbers in completely factored form:
100 = 2 · 2 · 5 · 5

18 = 2 · 3 · 3

29 = 29       (29 is a prime number)

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Greatest Common Factor (GCF) of two numbers

If we look at two or more numbers then they will have factors in common. For example
the factors of 40 are 1, 2, 4, 5, 8, 10, 20 and 40,

and the factors of 50 are 1, 2, 5, 10, 25 and 50.

We have shown the common factors in red. The greatest common factor is the largest of all the common factors. The greatest common factor of 40 and 50 is 10. Here are some more examples of greatest common factors:
the GCF of 24 and 30 is 6

the GCF of 24, 30 and 33 is 3

the GCF of 7 and 21 is 7

the GCF of 7 and 13 is 1


Here is a procedure to find the greatest common factor of two or more numbers. We illustrate with the numbers 24 and 30. Factor the numbers completely and line up their factors. (By this we mean put common factors below each other and when either number is missing a factor then leave a space for it.)
Now it is easy to see the factors that both numbers have in common. Because they both have 2 and 3 in common the greatest common factor must be 2 · 3 = 6.




Least Common Multiple (LCM) of two numbers

The multiples of a number are the numbers that have that number as a factor. For example the multiples of 5 are 5, 10, 15, 20, 25, …

If we look at two or more numbers then they will have multiples in common. For example
the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, …

and the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, …
We have shown the common multiples of 3 and 4 in red. They are 12, 24, … The smallest of the common multiples is called the least common multiple. Here are some more examples of least common multiples:
the LCM of 9 and 20 is 180

the LCM of 2, 3 and 5 is 30

the LCM of 7 and 21 is 21

Here is a procedure to find the least common multiple of two or more numbers. We illustrate with the numbers 24 and 30. Factor the numbers completely and line up their factors. (By this we mean put common factors below each other and when either number is missing a factor then leave a space for it.)
The least common multiple must contain all the factors that are in either one number or the other, but the factors are not used twice when they are common to both numbers. Lined up like this it is easy to spot the common factors. The least common multiple must be 2 · 3 · 2 ·  2 · 5 = 120.

You often need to find the factors that each of the original numbers must be multiplied by to give the LCM. Lined up like this they are easy to spot. They are just the missing factors. In this example 24 must be multiplied by the missing 5, and 30 must be multiplied by the missing 2 · 2 or 4.