If you are looking for the factors of 30, you are in the right place!
In this lesson, you will learn what they are, how to find them, and lots of other fun factor facts.
Before we start looking at 30, it is essential to understand what a factor is.
Very simply, a factor is one number that goes into another number exactly.
For example, 2 is a factor of 4 because it goes into 4 without leaving a remainder.
Each number that goes exactly into 30 is also known as a divisor. Now let’s have a look at what the factors are and how we find each of them.
All Factors of 30
The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.
30’s proper factors are 2, 3, 5, 6, 10 and 15.
Proper factors simply means any other than 1 or the number itself.
What are the Factor Pairs of 30?
A factor pair are two numbers that multiply together to make the original number.
In the case of 30, we are looking for all pairs of integers that multiple together to make 30, and here they all are:
Notice that we also included pairs of negative numbers, since two negative numbers multiply together to give a positive number. They are factor pairs of 30 too!
We could also add (6, 5), (10, 3), etc., to the list, but multiplication is commutative, meaning the order does not matter – you get the same result either way.
Looking at the list of factor pairs, we get the following list of 8 positive numbers, also called divisors, of 30:
They are also called divisors because these are the only numbers that give a whole number answer when dividing 30.
6 is a whole number, so both 5 and 6 are factors.
7.5 is not a whole numbers, so 4 is not a factor.
Bear this fact in mind. It will be useful when you are finding the factors of a number.
Prime Factors of 30
A prime is a number that is divisible by itself and 1.
For example, 8 is not prime as it is divisible by 4 and 2.
However, 2 is prime.
The first 15 prime numbers are:
It turns out that there are infinitely many prime numbers – this list goes on and on. Forever!
What is the largest prime number you can come up with? Can you check that it is prime?
The prime factors of 30 are just the factors that are also prime. By looking at the two lists above, we see that for 30, they are 2, 3, and 5.
You will see why these prime factors are important in The Prime Factorization of 30.
How to Find the Factors of 30
One simple way to find factors is to ‘guess and check’: guess a number, then check whether it divides 30 and gives a whole number.
Don’t guess randomly though, be systematic.
30 ÷ 2 = 15
30 ÷ 3 = 10
30 ÷ 4 = 7.5
30 ÷ 5 = 6
We don’t go any further because dividing 30 by 6 gives 5 – that same factor pair comes from dividing by 5.
Only dividing by 4 gave a result that was not a whole number.
From that, we have found that 1, 2, 3, 5, 6, 10, 15 and 30 are all factors of 30.
Notice that the pairs of numbers are exactly the factor pairs listed in What are the Factor Pairs of 30?
Another way to look at it is that the smallest number in a factor pair of 30 must be smaller than the square root of 30; otherwise, the product of the two numbers would end up being bigger than 30.
The square root of 30 is 5.477… so we know that we only need to check the numbers up to 5.
As you’ve seen, the ‘guess-and-check’ method of finding factors works quite well, as long as the number’s square root isn’t too large!
Now, here’s one for you to do yourself.
They are 1, 2, 4, 5, 8, 10, 20, and 40.
The Prime Factorization of 30
Remember what the prime factors of 30 are? They were 2, 3, and 5. We can see how 30 breaks down into a product of its prime factors using a factor tree:
First, split 30 up into one of its factor pairs. Here are both (2, 15) and (3, 10).
Then, continue this process with the resulting factors, splitting them up into pairs of smaller factors, unless it is prime, in which case it won’t split up any further.
What you are left with is called the prime factorization of 30:
Looking at these two different ‘trees’, you can see that it doesn’t matter how you split up the factors, it’s always the same result.
It turns out that prime factorizations are unique! This means that every number can be written as a product of its prime factors in only one way.
If you used a factor tree to work it out, it might have looked something like this?
Although don’t worry if you chose different factors to start with, the result will always be the same.
Why isn’t 1 Prime?
Have another look at the definition of a prime number that was given earlier:
Numbers are called prime if their only factors are one and themselves.
Doesn’t 1 fit that definition?
Even though the only factor of 1 is 1, which is both 1 and itself, mathematicians exclude 1 from the ‘club’ of primes because it is quite helpful for it not to be one!
If you allow 1 to be a prime, you can no longer say that prime factorizations are unique! Looking at 30 as an example:
= 2 × 3 × 5 × 1
= 2 × 3 × 5 × 1 × 1
You can just keep on adding 1’s to the prime factorization without changing the result, so the uniqueness is gone!
This is why 1 is not considered a prime.
Isn’t 30 Interesting?
Where do we use the number thirty?
- The months April, June, September, and November each have 30 days. In fact, the average length of a month in days is slightly over 30, at approximately 30.45 days.
- There are a total of 30 major and minor keys in music: a major and minor version of each of the 15 different key signatures.
- You must be at least 30 years old to become a senator in the Senate of the United States. You also need to have been a US citizen for at least 9 years, and you need to live in the state that you want to become a senator for. President Joe Biden was only 29 when he was elected to become a senator! But he didn’t take the oath of office until he turned 30 in the next year, 1973.
- 30 different teams compete in both Major League Baseball and the National Basketball Association.
30 has some interesting mathematical properties too!
30 is a square pyramidal number, meaning if you have 30 blocks, you can arrange them into a square pyramid:
As you can see in this diagram, it’s the fourth square pyramidal number.
The fifth square pyramidal number is 55
There are 30 partitions of the number 9. A partition shows a number as the sum of smaller numbers.
For example, there are 7 different partitions for 5:
2) 4 + 1
3) 3 + 2
4) 3 + 1 + 1
5) 2 + 2 + 1
6) 2 + 1 + 1 + 1
7) 1 + 1 + 1 + 1 + 1
They are as follows:
2) 6 + 1
3) 5 + 2
4) 5 + 1 + 1
5) 4 + 3
6) 4 + 2 + 1
7) 4 + 1 + 1
8) 3 + 3 + 1
9) 3 + 2 + 2
10) 3 + 2 + 1 + 1
11) 3 + 1 + 1 + 1 + 1
12) 2 + 2 + 2 + 1
13) 2 + 2 + 1 + 1 + 1
14) 2 + 1 + 1 + 1 + 1 + 1
15) 1 + 1 + 1 + 1 + 1 + 1 + 1
Phew! Good work if you found them all.
The number 30 itself has tonnes of partitions: 5604 to be exact! We’ll not try and list them here!
To Sum Up (Pun Intended!)
We found the factors of 30 by testing to see whether a potential factor gave a whole number when it divides 30. Those which did are the factors:
We also found the prime factorization of 30:
Make sure that you understand all the methods used to find the factors and the prime factorization. If you’re not sure, have a go at factoring some other numbers, using the same methods that we used here.
Leave a comment below if you have solutions to any of the questions in the article, or if you have any of your own interesting facts about the number 30, or anything else we discussed, that you want to share – related to math or not, we’d love to hear them!
If you want a quick reference, look at this useful table of the factors of 1 to 100, including primes, and the first 20 multiples.