40 is a really nice number to work with. It feels like a natural choice for exploring a new topic in math.
In this lesson, we will break down 40 to find its various types of factors, look at their properties, and apply the method we used to be able to factorize any number.
You might be wondering what factors are? Before we jump in and find the factors of 40, you should understand their definition.
“A factor is a number which divides another number exactly, leaving no remainder.”
This definition uses division. It might help to think of it the other way around, using multiplication:
“When multiplied together, two factors make the original number exactly.”
All Factors of 40
40 has 8 factors: 1, 2, 4, 5, 8, 10, 20, 40.
Adding the factors up, the total is 90.
40’s proper factors are all its factors except 1 and 40.
There are 6 proper factors for 40: 2, 4, 5, 8, 10, 20. If we also add these up, the total is 49.
40’s factors give it a special property which makes it abundant. Numbers are called abundant if the sum of their proper factors or half the sum of all their factors is bigger than them.
Both conditions are true for 40, so 40 is abundant.
Factor Pairs of 40
The multiplication definition of factors says that two factors can be multiplied to make the original number exactly.
Two factors that multiply together to make the original number are called a factor pair. If you have one factor, find its partner by dividing the original number by the one you do have.
4 is a factor of 40.
We can find its pair by dividing 40 by 4, giving the factor pair:
When the pair of factors is multiplied, we have:
In most cases, the factors in a pair are different from each other. This means that most numbers have an even number of factors – each pair adds two factors to the total.
If the original number is square, its square root is a factor. The factor pair of a square root is…the square root again – it’s in a pair with itself!
For example, 100 is a square number, so 10 is in a pair with another 10! Because 10×10=100.
You wouldn’t list 10 twice, so square numbers have an odd amount of divisors. Numbers that aren’t square have an even amount of factors.
40 is not a square number so it has an even number of factors – as you’ve seen! All its factors can be matched into pairs:
40 = 20 × 2
40 = 10 × 4
40 = 8 × 5
What if we go off the beaten track and include negative factors?
Allowing negative numbers to be factors doubles the total! In these equations, making all the factors negative would still give the correct answer of 40 every time.
The factor pairings are the same. Instead of pairing positive factors together, negative factors are paired together. The factors in each pair’s factors must have the same sign, or else they won’t multiply to give a positive original number.
40 = (-20) × (-2)
40 = (-10) × (-4)
40 = (-8) × (-5)
Prime Factors of 40
Prime factors, as you may guess, are factors that are prime numbers.
A prime number is one that cannot be divided by anything other than 1 or itself.
The factors are 1, 2, 4, 5, 8, 10, 20 and 40.
Then the prime factors are 2 and 5.
A rule to remember for later is that all numbers can be written as products of their prime factors.
We will learn how to find the prime factorization below.
We’ve just seen that every number has at least 1 and itself as a factor. For this reason, these numbers don’t tell us anything unique or interesting about the number.
It’s often useful to ignore 1 and the original number from the list of factors for this reason.
Proper factors are a special subcategory of “regular” factors.
They are all the factors except 1 and the number itself.
The proper factors of 40 are: 2, 4, 5, 8, 10 and 20.
Now you know 40’s factor secrets!
It’s no good just having the answers – it’s important to be able to get them.
We will now work out how to factorize 40 and then work through a more challenging example to boost your confidence.
The easiest way to find the factors of a number is to use a calculator. There are lots of possible factors to check, and a calculator will save lots of time!
Failing that, or if you’re in an exam where you can’t use a calculator, use the following divisibility rules.
Divide the number you want to factorize by each number, starting from 1 and counting up, to check which numbers are factors.
|Divisible by …?||Test|
|1||No test needed – all numbers are divisible by 1|
|2||Even – number ends in 2, 4, 6, 8, or 0|
|3||Sum of digits is a multiple of 3|
|4||The number made by the last 2 digits are divisible by 4|
|5||Number ends in 0 or 5|
|6||Divisible by 2 and 3|
|7||No simple test!|
|8||Divisible by 4 after being halved|
|9||Sum of digits is a multiple of 9|
|10||Number ends in 0|
If all else fails, there is always long division.
Now, every time you find a number that divides your number exactly, write it down – it’s a factor. Then find its pair by dividing the original number.
Stop searching for factors once the number you’re dividing by becomes equal to or bigger than the square root of 40, which is 6.324. So you can stop checking at 6.
At this point, you have already found all the factors! If you kept looking, any large factor found would already be on the list in a pair with a smaller factor.
How to Find the Factors of 40
We will apply this method to 40. Though you may find the factors in your head, and the process may seem tedious, it’s important to understand and practice the technique.
It will become second nature with some practice – this method finds all the factors and is pretty fast!
1 is our first factor because all integers are divisible by 1
40 ÷ 1 = 40
1’s pair is (1, 40)
40 is even, so it is divisible by 2.
40 ÷ 2 = 20
2’s pair is (2, 20)
Adding 40’s digits gives 4+0=4, and 4⁄3=1.333 so 40 is not divisible by 3
40 ÷ 3 = 13.33…
40’s last two digits are just 40! Luckily, it’s easy to see that 4 is a factor
40 ÷ 4 = 10
4’s pair is (4, 10)
40’s last digit is 0, so it’s divisible by 5
40 ÷ 5 = 8
5’s pair is (5, 8)
40 is not divisible by 6 because it isn’t divisible by 3
40 ÷ 6 = 6.66…
We don’t need to check any for any factors higher than 6 because 7 is larger than the square root of 40.
We have found all of 40’s factors: 1, 2, 4, 5, 8, 10, 20, 40.
Prime Factorization of 40
Breaking numbers down into their prime factors lets us see “inside them” – the factors are their building blocks, letting us find relationships between numbers where we might not expect it.
Multiplying these prime factors together will always give the original number.
To find a number’s prime factorization, you must first factorize it. Then, if the factors are not prime, factorize these factors. This process is repeated until all the factors are prime!
The easiest problems are when the original number is prime because there’s no work required! Prime numbers are already their own prime factor – they cannot be factorized or written more simply.
1. Don’t include 1 in the factorization because it doesn’t tell you anything interesting – 1 is a factor of every number!
2. If the same factor appears more than once, you must include each repeated copy. If you don’t, the product of the factors won’t be the original number.
Finding the prime factorization of 40 starts with a choice. Pick any pair of its proper factors and the factorization will end up being the same.
If either or both factors in the pair are not prime, factorize them. If either or both factors are not prime, factorize these factors!
We like to keep things neat, so we will start with the pair with the smallest prime factor in it. You can start with whichever factor pair you like.
Then, keep breaking the factors of 40 down, looking for primes, until the process cannot be repeated any more, and only prime factors remain.
The smallest prime number that goes into 40 is 2, so let’s break 40 into the pair (2, 20)
2 is obviously prime, so it is left alone. 20 isn’t though, so factorize it again. The smallest prime number that goes into 20 is 2 again, so…
We need to do the same for 10, and again it’s smallest prime factor is 2:
2 and 5 are both prime so we are finished. Substitute our factorizations for 20 and 10 into the first factorization for 40.
This gives us a single equation, the prime factorization of 40:
If we tidy this up, it looks like this:
It might help to visualize this process as a tree. When all the factors have been factorized, we collect the prime factors from the ends of branches.
Ready to test what you have just learned? Here’s something more challenging.
Let’s look at 525, a bigger number with a more complex factorization. The process is the same as before!
1 is a factor. 1’s pair is (1, 525)
2 is not a factor because 525 is odd.
3 is a factor because 5+2+5=12 and 12÷3=4. 3’s pair is (3, 175).
Looking at the last two digits, we see that 25 isn’t divisible by 4, so 525 isn’t either.
5 is a factor because the last digit is 5. 5’s pair is (5, 105).
6 isn’t a factor because 525 isn’t divisible by 2.
7 is a factor. 7’s pair is (7, 75).
There aren’t any more factors until 15! 15’s pair is (15, 35).
Now, there aren’t any factors until 21! 21’s pair is (21, 25).
22 isn’t a factor because 525÷22=22.86, which is not a whole number.
We stop checking here because is 23 is bigger than the square root of 525, 22.9.
All factors bigger than 22 have already been calculated as the pairs of factors smaller than 22.
So, 525 has 12 factors: 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525.
Well done! Now, try the next step on your own.
The smallest prime that 525 is divisible by is 3.
175 is not prime so factorize it with its smallest prime, 5:
35 is not prime, so it is factorized with its smallest prime, 5:
5 and 7 are both prime so we are done! That was surprisingly quick.
Like before, we substitute our non-prime factors with their prime factorization.
Write repeated factors as exponents to tidy up:
This can be pictured by collecting the prime factors at the ends of the tree diagrams branches.
Isn’t 40 Interesting?
40 is used frequently in the real world. Next time you come across it, think back to how it’s factorized and see if you can find its prime factors again! Here are some fun facts about 40:
- There are 40 spaces on a regular Monopoly board, but it always seems like more when you’re losing!
- Writing out 40 as forty, you can see that its letters are in alphabetical order. Forty is the only number which this is true for!
- -40 is the only temperature which is the same in both Fahrenheit and Celcius. To convert from Fahrenheit to Celcius, you must subtract 32, then multiply by 5⁄9.
On top of being a useful example in this lesson, 40 has many special properties in math. It turns up all over the place:
- 40 is a Harshad number. This fancy name means that its digits add up to one of its factors: 4+0=4 and 4 is a factor.
- 40 is the 4th pentagonal pyramidal number. These numbers represent how many objects fit inside pyramids with a pentagon base.
- 40 is a semiperfect number, meaning that some of its proper factors can be added up to make 40. Why isn’t it perfect, I hear you ask? It’s the sum of some of the proper factors, not all of them:
To Sum Up (Pun Intended!)
Today, you learned the definition of factors, proper factors, and prime factors.
We stated the prime factorization and factors of 40, understood a thorough technique to find any number’s factors, then used it on 40 to derive its factors and prime factorization.
When the starting number is big, factoring can be a tedious process.
It’s easiest with a calculator, but divisibility rules help if you don’t have one.
Starting with 1 and counting up, you must check if the number is divisible and find its pair if it is. The search can stop when you reach the square root.
Prime factorization is all about breaking a number down. Factorize it and, if its factors are not prime, factorize the factors! This process is repeated until all the factors are prime.
The next step is to practice. Try repeating the factoring process on other numbers you like! If you have any questions or a good 40-fact, please comment below.
We hope you understand the various types of factors and feel confident enough to find them!