LCM and HCF

Greatest Common Factor

The highest number that divides exactly into two or more numbers.
It is the "greatest" thing for simplifying fractions!

Let's start with an Example ... 

Greatest Common Factor of 12 and 16

1. Find all the Factors of each number,
2. Circle the Common factors,
3. Choose the Greatest of those

So ... what is a "Factor" ?

Factors are the numbers we multiply together to get another number:

A number can have many factors:

Factors of 12 are 1, 2, 3, 4, 6 and 12 ...

... because 2 × 6 = 12, or 4 × 3 = 12, or 1 × 12 = 12.

(Read how to find All the Factors of a Number. In our case we don't need the negative ones.)

Why is this Useful?

One of the most useful things is when we want to simplify a fraction:

Example: How could we simplify 1230 ?

Finding the Greatest Common Factor

Example:

Two Numbers	Factors	Common Factors	Greatest Common Factor	Example Simplified Fraction
9 and 12	9: 1,3,9 12: 1,2,3,4,6,12	1,3	3	912 = 34

Other Names

The "Greatest Common Factor" is often abbreviated to "GCF", and is also known as:

• the "Greatest Common Divisor (GCD)", or
• the "Highest Common Factor (HCF)"

COMPARING FRACTIONS

Comparing Fractions

Sometimes we need to compare two fractions to discover which is larger or smaller. There are two easy ways to compare fractions: using decimals, or using the same denominator.

The Decimal Method of Comparing Fractions

Just convert each fraction to decimals, and then compare the decimals.

Example: which is bigger: 38 or 512?

Convert each fraction to a decimal.

We can use a calculator (3÷8 and 5÷12), or the method on Converting Fractions to Decimals.

Anyway, these are the answers I get:

38 = 0.375, and 512 = 0.4166...

So 512 is bigger.

The Same Denominator Method

The denominator is the bottom number in a fraction.

It shows how many equal parts the item is divided into

When two fractions have the same denominator they are easy to compare:

Example:

49 is less than 59 (because 4 is less than 5)

But when the denominators are not the same we need to make them the same (using Equivalent Fractions).

Example: Which is larger: 38 or 512?

Look at this:

• When we multiply 8 × 3 we get 24,
• and when we multiply 12 × 2 we also get 24,

so let's try that (important: what we do to the bottom we must also do to the top):

× 3 3  =  9 8 24 × 3

and

× 2 5  =  10 12 24 × 2

We can see that 924 is smaller than 1024 (because 9 is smaller than 10).

so 512 is the larger fraction.

Making the Denominators the Same

There are two main methods to make the denominator the same:

• Common Denominator Method, or the
• Least Common Denominator Method

They both work, use which one you prefer!

Example: Which is larger: 56 or 1315 ?

Using the Common Denominator method  we multiply each fraction by the denominator of the other:

× 15 5  =  75 6 90 × 15

and

× 6 13  =  78 15 90 × 6

We can see that 7890 is the larger fraction

so 1315 is the larger fraction.

Least Common Multiple(LCM)

The smallest positive number that is a multiple of two or more numbers.

Let's start with an Example ... 

Least Common Multiple of 3 and 5:

List the Multiples of each number,

The multiples of 3 are 3, 6, 9, 12, 15, 18, ... etc
The multiples of 5 are 5, 10, 15, 20, 25, ... etc

Find the first Common (same) value:

The Least Common Multiple of 3 and 5 is 15

( 15 is a common multiple of 3 and 5, and is the smallest, or least, common multiple )

What Did We Do?

The trick was to list the multiples of each denominator, then find the Least Common Multiple

In the previous example the Least Common Multiple of 3 and 6 was 6.

In other words the Least Common Denominator of 13 and 16 is 6.

Here are the steps to follow:

Find the Least Common Multiple of the denominators (which is called the Least Common Denominator). Change each fraction (using equivalent fractions) to make their denominators the same as the least common denominator Then add (or subtract) the fractions, as we wish!

Example: What is 16 + 715 ?

The Denominators are 6 and 15:

multiples of 6:		6, 12, 18, 24, 30, 36, ...
multiples 15:		15, 30, 45, 60, ...

So the Least Common Multiple of 6 and 15 is 30.

Now let's try to make the denominators the same.

Note: what we do to the bottom of the fraction,

we must also do to the top

When we multiply 6 × 5 we get 30, and when we multiply 15 × 2 we also get 30:

× 5 1  =  5 6 30 × 5

and

× 2 7  =  14 15 30 × 2

Now we can do the addition by adding the top numbers:

530 + 1430 = 1930

The fraction is already as simple as it can be, so that is the answer.