Working with fractions: How to find the Least Common Multiple or the Greatest Common Factor?

Check your skills on fractions with this quiz. If you score below 70%, you probably need to do a little more revision first! Read until the end of this post and retry the quiz.

Fractions, as you probably already know, are numbers that represent part of a whole, e.g. 1/2 (one half) or 2/5 (two fifths). The number above the slash (/) is known as the numerator, and the number below is known as the denominator. The objective of this post is to revise a few basic concepts of calculating fractions before attempting the quiz which follows. If you are unfamiliar with maths terminology, here's an excellent interactive dictionary.

To multiply fractions, we just need to multiply the numerators together, and the denominators together, then reduce the resulting fraction:

5/6 x 5/8 = 25/48

4/5 x 5/6
= 20/30
= 2/3

To divide fractions, turn the fraction you are dividing by (divisor) upside down, then multiply:

3/8 ÷ 3/4
= 3/8 x 4/3
= 12/24
= 1/2

3/4 ÷ 3
= 3/4 x 1/3
= 3/12
= 1/4

To add or subtract fractions with the same denominator, we perform the arithmetical operation on just the numerator, and leave the denominator as it is:

3/8 + 7/8
= 10/8
= 1 2/8
= 1 1/4

7/8 - 3/8
= 4/8
= 1/2

However, if the denominators are different, we must first convert the fractions into equivalent fractions with the same denominators. To do this, it is useful to find the lowest common multiple of the denominators.

BASIC METHOD

The least common multiple (LCM) (also known as lowest common multiple or smallest common multiple) of two numbers is the smallest number that is a multiple of both of them. A multiple of a number can be divided into the number without a remainder.

For example,

multiples of 4 are:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ...

(add 4 to each to get the next multiple).

Multiples of 6 are:

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, ...

(add 6 to each to get the next multiple).

Common multiples of 4 and 6, therefore, are numbers that are found in both lists:

12, 24, 36, 48, ....

The least common multiple, as you can see, is 12.

3/4 + 5/6
= 9/12 + 10/12
= 19/12
= 1 7/12

This is the very basic method of calculating the LCM. However, if you need to calculate the LCM of 2 big numbers, this method is not very convenient. So, we look at another way of calculating it. Let us find the LCM of 18 and 54.

VENN DIAGRAM

First, we need to find the prime factors of these 2 numbers. Prime factors of a number are the prime numbers that divide into that number exactly, without leaving a remainder. The process of finding these numbers is called prime factorization:

18 = 2 x 3 x 3
54 = 2 x 3 x 3 x 3

We now draw a Venn diagram, which is a diagram of two circles intersecting another. We write all the factors that these two numbers have in common in the intersection (2, 3, 3). We write the unique factors of 18 on the left circle (none) and those of 54 on the right (3).

To find the least common multiple, multiply all the numbers we see in the diagram: 2 x 3 x 3 x 3, which gives 54.

To calculate the greatest common factor (also known as the highest common factor or the greatest common divisor), we multiply only the numbers in the intersection: 2 x 3 x 3, which gives 18.

Here's another example. Let us take a look at the numbers 48 and 180. Breaking them down, we have:

  48 = 2 x 2 x 2 x 2 x 3
180 = 2 x 2 x 3 x 3 x 5

We see that the common factors of these two numbers are 2, 2, and 3.
The resulting Venn diagram is as such:

The least common multiple is, therefore, 2 x 2 x 2 x 2 x 3 x 3 x 5, which equals 720.
The highest common multiple is 2 x 2 x 3, which equals 12.

The highest common multiple is useful for simplifying fractions, e.g. 48/180. Dividing both the numerator and the denominator by 12 will give the answer 4/15.

Example: 3/48 + 15/180

To be able to calculate this, we need to convert these fractions to have a common denominator, which we know from our previous calculation to be 720:

So, we divide 720 by 48 (the denominator), which gives us 15. We then multiply this by the numerator, 3, to give the answer 45.

3/48 = 45/720

Likewise, we divide 720 by the denominator, 180, to give 4. Multiplying this by the numerator, 15, we get 60:

15/180 = 60/720

3/48 + 15/180
= 45/720 + 60/720
= 105/720