180 is a great number to study. Not only does it have lots of nice mathematical properties, but it’s also fun to shout out if you’re playing darts!
In this lesson, we will break 180 down into factors, look at their properties, then learn to use the techniques to factorize any number.
Before you get stuck in, let’s clear something up. What does factor mean?
Factors are numbers that divide another number exactly, without leaving a remainder.
It might be easier to think of this definition the other way around – two factors can be multiplied to make the original number exactly.
If you’re looking for a quick reference, here are all the factors and a bunch of multiples for 1-100!
All Factors of 180
The factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90 and 180.
When thinking about factors, you should assume that they are positive, however, they can sometimes be negative. We’ll look more at this later.
Factor Pairs of 180
Every factor has a pair, which it multiplies with to make the original number.
You can find the second factor in each pair by dividing the original number by the first factor in the pair. This means that when you multiply the pair together, you always get the original number.
We know that 10 is a factor of 180, so to work out the other number in the pair, you need to divide 180 by 10, which is 18!
So both 10 and 18 make a factor pair of 180.
You can tell that 180 is not a square number because it has an even number of factors. This means all its factors can be matched into pairs which multiply to make 180.
The factor pairs of 180 are:
But, what would happen if we bent the rules?
If you also think about negative numbers, the number of possible factors doubles! You could make each factor negative and these equations would still make sense.
Don’t forget that two negative numbers multiply to make a positive number, but a positive multiplied by a negative is negative! The factors in each pair must have the same sign.
Prime Factors of 180
Prime factors, as their name implies, are factors that are prime numbers.
For a number to be a prime number, only itself and 1 can go into it with no remainder.
11 is prime because it is only divisible by 11 and 1.
There are no other numbers that go into 11 with no remainder.
A number that is not prime is called a composite number.
4 can be divided by 2, so it is a composite number.
For 180, the number we are focusing on in this lesson, there are prime factors 5, 3, and 2.
An important rule to remember is that every number can be written as a product of its prime factors.
This is known as prime factorization, which we will take a look at later on in this lesson.
A number’s proper factors are all its factors except 1 or itself. They are a subcategory of the “regular” factors defined above.
The number 180’s proper factors are 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60 and 90.
24 has 8 factors: 1, 2, 3, 4, 6, 8, 12, 24. It has 6 proper factors: 2, 3, 4, 6, 8, 12.
27 has 4 factors: 1, 3, 9, 27. It has 2 proper factors: 3, 9.
89 has 2 factors: 1, 89. It doesn’t have any proper factors.
Can you see a pattern?
Every number has two fewer proper factors than “regular” factors. Prime numbers, including 89, always have two factors so they never have any proper factors.
If you prefer to read, continue here.
You have the list of 180’s factors and prime factors. Now that you know the destination, go on a factorization journey to get there!
We will also look at a harder example to give you the confidence to solve problems alone.
The easiest way to find the factors of a number is to either use the following divisibility rules or a calculator.
To avoid missing out any factors, you might find it easier to start from 1 and work upwards.
Divisibility rules are a handy set of tricks, or shortcuts, that let us see quickly and easily whether a large number can be divided by a smaller one.
These are so useful in fact, that we made a printable table and worksheet for the divisibility rules from 2 to 15, for you to keep with you. Just click or tap the images below to download them!
If you have a calculator, you can simply divide the number by 1, 2, 3, etc. to see if you get a whole number!
Each time you find a number that goes into the main number, write it down – it is a factor. You need to find the pair, by dividing the main number by the new factor you’ve just found.
Once the number you are checking becomes larger than the square root, you can stop searching.
All the factors beyond this point have already been found – they are the pairs of the smaller factors you already have!
How to Find the Factors of 180
Let’s apply the divisibility rules to 180.
Even though you might be able to find the factors in your head, it is important to practice the method on smaller numbers so you can use it confidently on more challenging numbers!
1: 1 is the first factor because all integers are divisible by 1
2: 180 is even so it is divisible by 2.
3: Adding 180’s digits together gives 1+8+0=9 and 9÷3=3, so 180 is divisible by 3
4: Looking at 180’s last two digits, we see 80 is divisible by 4, so 180 is divisible by 4
5: The last digit of 180 is 0 so it is divisible by 5
6: 180 is divisible by 6 because is divisible by both 2 and 3
7: 7 does not go into 180.
8: Nor is it divisible by 8, because half of 180 is 90, and 90 is not divisible by 4.
9: Adding 180’s digits equal 9, and 9÷9=1 so 180 is divisible by 9.
10: 180’s last digit is 0 so it is divisible by 10
11: It is not divisible by 11
12: 180 is divisible by 3 and 4 so is divisible by 12
13: 13 does not go into 180.
The square root of 180 is 13.416… so we don’t need to check any more numbers.
Now you have found that 1, 2, 3, 4, 5, 6, 9, 10, and 12 are all factors of 180, you need to find the pairs of each of those numbers.
To do that, divide 180 by each factor!
180 ÷ 2 = 90
180 ÷ 3 = 60
180 ÷ 4 = 45
180 ÷ 5 = 36
180 ÷ 6 = 30
180 ÷ 9 = 20
180 ÷ 10 = 18
180 ÷ 12 = 15
The answers to these sums are the rest of the factors!
So the complete list of 180’s factors is: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.
Prime Factorization of 180
The neatest way to find a number’s prime factorization is to divide by the smallest prime factor. Then, repeat this process, breaking the factors down until there are no composite numbers left!
A composite number is one that can be divided by something other itself or 1 with no remainder. This means it is NOT a prime number.
It is helpful to visualize the process as an upside-down tree because, in the end, we collect the circled prime factors from the ends of the branches.
Start by dividing 180 by its smallest prime factor, 2.
2 is prime so it is circled and stays as it is. 90 is not prime, and so is broken down further.
Identify its smallest prime factor. For 90, it is easy to see it’s also 2 because 90 is even.
45’s smallest prime factor can’t be 2, so you need to check for divisibility by each prime number. The next smallest is 3. Use the divisibility rule for 3 by adding 45’s digits together.
9 is divisible by 3, so 45 is divisible by 3 as well.
15 is not a prime number, so you’re not done just yet. It is not divisible by 2, so again, check the next smallest prime, 3. Yes! It is easy to see that 3 divides 15.
5 is also a prime number, so you are finished! Here is the final factor tree.
Gather up all the prime numbers and write them as a multiplication:
This is more simply written with exponents, or powers:
This is the prime factorization of 180!
Up for a challenge? Let’s look at a bigger number, 288. You can use the same method as before to find its factors. It’s much easier with a calculator when there are so many numbers to check!
See if you can work out the answer and then check if you’re right!
The factors of 288 are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, and 288.
Now that you have its factors, using the factor tree method above, you can tackle this…
To tidy things up, write the repeated terms as exponents:
180 is a Mathematically Special Number
On top of having a huge list of factors, 180 has several special properties. These are useful in number theory, an area of math you might be interested in.
180 is a Harshad number. This means that the sum of its digits is one of its factors:
and 9 is a factor.
180 has more factors than any smaller positive integers so it is called a highly composite number.
It is the sum of two squares: 180=62+122. This means that a right-angle triangle with short sides 6 and 12 would have a hypotenuse the same length as the square root of 180.
180 can be written as sums of consecutive prime numbers in two different ways:
The first is:
And the second way is:
To Sum Up (Pun Intended!)
In this lesson, we defined factors, proper factors, and prime factors.
We stated the factors of 180, learned a fool-proof method to find the factors of any number, then applied it to find 180’s factors and prime factorization.
Factoring can be a long process when the number is big. Starting with 1 and counting up, check if the number is divisible and find the factor’s pair if it is. You can stop searching for factors when you reach the square root of the number!
To find a number’s prime factorization, you factorize it. If its factors are not prime, you factorize the factors! This process continues until you reach prime factors.
We hope you feel confident to factor challenging numbers and understand the definitions of various factor types. The next step is to practice more! If you have any questions or a good fact about 180, please leave a comment.