In this lesson, you’ll learn:
- How to find the factors of 18
- What factors pairs are
- How to find the prime factorization
A factor is a number that neatly divides another number, leaving no remainder or decimal.
Finding factors is really simple once you know how to!
By the time you’ve finished, you’ll be able to find the factors of any number you like!
All Factors of 18
The factors of 18 are:
Factor Pairs of 18
Factor pairs are combinations of factors that multiply to give the original.
Square numbers have an odd number of pairs because the square root is happy little pair by itself, but all others have an even number.
18’s factors can be sorted into pairs. When multiplied together, these pairs give 18.
18 has 6 factors and 3 factor pairs:
This is a sneaky hint at how we can find a number’s factors in the first place. Have a think about how factor pairs could be helpful in our technique!
One of the rules of factors is that they are positive numbers. This is because including negative integers doesn’t tell us anything new – we just get the negative version of the same numbers.
What if we did include them?
So (-1, -18) is a pair.
Following on from that, the numbers in every pair we had before can be swapped with their negative counterparts:
When negative factors are allowed, the number of factors doubles.
Prime Factors of 18
Prime numbers are special because they are only divisible by 1 and themselves. They always have two factors and never have proper factors.
The idea of prime numbers and factors can be combined into prime factors. As the name says, these are factors that are prime numbers.
18’s prime factors are:
Think of these as their basic building blocks! You will learn how to find the prime factorization of 18, using the building blocks in just a moment.
The same method can be used to factor every number. It might look tedious but is quick with practice and a calculator!
Step 1) Write down 1 and the original number. These are always factors so are a good place to start!
Step 2) Is the number divisible by 2? If it is, write down 2 and find its factor pair by calculating the original number, divided by the first factor.
Step 3) Repeat step 2, checking if the original number is divisible by 3 and finding its factor pair if it is.
Step 4) Keep counting up in this sequence, checking if each number is a factor and finding its pair if it is. Stop when you reach the square root of the original number.
You can stop there because every factor bigger than the square root is in a pair with a factor smaller than the square root.
All the smaller factors have been found and their pairs, the bigger factors, have been found too!
There are lots of divisions to calculate. If you don’t have a calculator, you can use the following divisibility rules or calculate some divisions by hand.
All you really need to factorize a number is brainpower and some patience!
1) All integers are divisible by 1
2) All even numbers are divisible by 2
3) Add up the digits. If the sum is divisible by 3, the original number also is
4) Just look at the last two digits. If this number is divisible by 4, so is the original number
5) If the last digit is 0 or 5, the number is divisible by 5
6) If the number is divisible by 2 and 3 using the above rules, it is also divisible by 6
7) No easy divisibility test
8) If the last three digits, as their own number, are divisible by 8 then so is the original. For smaller numbers, divide the original number by 2 and test the result for divisibility by 4
9) Add up the digits. If the sum is divisible by 9, so is the original number
10) If the last digit is 0, the number is divisible by 10
How to Find the Factors of 18
Let’s work through this example together, using the divisibility rules. You can check that you find the same factors with a calculator!
1: 1 is always a factor.
1’s pair is:
2: 18 is even so 2 is a factor
2’s pair is:
3: 1+8=9 and 9 is divisible by 3, so 18 is too
3’s pair is:
4: Dividing 18 by 4 gives a decimal so 4 is not a factor.
We can stop here because 5 is bigger than the square root of 18.
You can find the stopping point by listing the square numbers and seeing which two you can sandwich the target number between. The first 6 square numbers are:
The square root of 18 will be bigger than 4 but less than 5, so you only need to check numbers up to 4.
Prime Factorization of 18
Prime factors are interesting because every integer can be written uniquely as a product of its prime factors.
This means that prime factors describe every number in a fundamental way and that no two numbers have the same prime factorization. Just like snowflakes!
Finding the prime factorization takes slightly longer. Make sure you understand regular factoring before trying this!
We start by picking a pair of proper factors. Go wild, choose whichever you like! The answer will always be the same.
If the factors are prime, they’re perfect and can be left alone. If one or both factors aren’t prime, find a pair of their proper factors and check if these are prime.
Keep factoring the factors until they are all prime!
The prime factors must then be combined so that their product makes the original number. To do this, substitute any non-prime factors with their prime factorization.
Factoring the factors can be visualized as a tree.
The prime factors are collected from the tips of the branches when the factoring is finished.
Your final answer should look like a multiplication. Any repeated factors should be tidied up using exponent notation.
Don’t forget to double-check your answer!
Start with a pair of proper factors, say (3, 6).
3 is prime but 6 is not. We must find a factor pair for 6 but can leave 3 alone.
3 and 2 are both prime so we are done factoring! Phew, that wasn’t too bad.
Substituting the prime factors of 6 into the original pair (3, 6) gives:
3 is a repeated factor so should be expressed as an exponent. Our final answer is:
Ready for something more challenging?
The [[square root of 225 is 15, so 15 is the last number you need to check. As 225 is a square number, it should have an odd number of factors.
1: always a factor.
1’s pair is (1, 225)
2: 225 is odd so 2 is not a factor
3: 2+2+5=9 which is a multiple of 3 so 3 is a factor
3’s pair is (3, 75)
4: the last digits are 25, which is not a multiple of 4 so 4 is not a factor.
5: 225’s last digit is 5, so 5 is a factor
5’s pair is (5, 45)
6: not a factor because 2 isn’t a factor
7: 22-2×5=12 but 12 isn’t a multiple of 7 so it isn’t a factor
8: 2 isn’t a factor so 8 cannot be either
9: 2+2+5=9 which is clearly divisible by 9, so it is a factor
9’s pair is (9, 25)
10: 225’s last digit isn’t 0 so 10 isn’t a factor
11, 12, 13, and 14 are not factors because none of 225⁄11, 225⁄12, 225⁄13, or 225⁄14 are integers
15 is a factor because 3 and 5 are factors
15’s pair is (15, 15)
As predicted, 225 has an odd number of factors: 1, 3, 5, 9, 15, 25, 45, 75, 225.
Isn’t 18 Interesting?
Need a study break? Here are some fun math facts about 18:
- 18 is the first number whose prime factorization has the form p×q2. I didn’t promise a total break from factoring, only from studying them!
- Hexagon? Pentagon? Dodecagon? Nope, an 18-sided polygon is called an octadecagon! It has so many sides, you could mistake it for a circle.
- 18 is also a Lucas number. The Lucas numbers form a sequence like the Fibonacci numbers, where the next number is the sum of the previous two in the sequence. The Lucas numbers are: 2, 1, 3, 4, 7, 11, 18, 29…
Need a break from math altogether? 18 crops up plenty of times in the real world. Think of factoring next time you come across it:
- In most countries, 18 is the age when you legally become an adult.
- The 18th territory to join the United States was Louisiana, known for its rich culture, delicious food, and sunny weather. Sign me up for a holiday to New Orleans!
- Sonnet 18 is a well-known poem by a very well-known author, Shakespeare. You might recognize the first line “Shall I compare thee to a summer’s day?” from literature class. I quite like poetry, but it’s not for everyone!
To Sum Up (Pun Intended!)
In this lesson, you started by learning the basics of the different types of factors.
Factors are numbers that exactly divide it, leaving no awkward decimals or remainders. Factors can be arranged into pairs that multiply together to give the original number.
A proper factor is just a sub-category of normal factors. All factors are proper except for 1 and the original number being factored.
The last type of factors you learned are prime factors. They’re easy to remember because their name is their definition!
We then listed our special number 18’s factors and developed a factoring technique for tackling any problem.
Is there another way you prefer to find factors? Have you managed to split a huge number down into its prime factor building blocks? Don’t be shy, show off your knowledge and leave a comment!