In this lesson, you are going to learn what the factors of 60 are, and the simple methods you can use to find these factors yourself.
Once you learn this, you will be able to use the same methods to find the factors of any number you want, which is a useful skill to have.
60 gets used a lot – every minute is 60 seconds, and every hour is 60 minutes! With more understanding of the numbers that make it up, you can use it more effectively in your life.
But first … what is a factor?
A factor is a whole number that divides another number perfectly, without leaving a remainder or a fractional part.
If you share sweets between people evenly, then the number of people and the number of sweets they each get are both factors of the number of sweets!
For example, if you have 60 sweets, you can share them fairly between 10 people. Each person would get 6 sweets, so 6 and 10 are both factors of 60.
All the Factors of 60
The factors of 60 are:
Because factors are related closely to division, you will sometimes hear them called divisors. Likewise, division is related to multiplication: a number is also a multiple of its factors.
6 and 10 are factors of 60, and 60 is a multiple of 6 and 10!
To make sure you’re familiar with both terms, both ‘factor’ and ‘divisor’ will be used throughout this article, even though they mean the same thing!
Factor Pairs of 60
When sharing the sweets out equally, the number of people and the number of sweets each person gets, are factors.
These two factors are called a factor pair, and they multiply together to make the original total number of sweets.
Here’s a list of all the factor pairs for 60:
(2, 30) (-2, -30)
(3, 20) (-3, -20)
(4, 15) (-4, -15)
(5, 12) (-5, -12)
(6, 10) (-6, -10)
Looking at all the numbers that appear in the factor pairs, you’ll see again that there are 12 factors of 60:
Sometimes 1 and 60 will be excluded from this list since every number has 1 and itself as a factor. The remaining factors are then called the proper factors:
Prime Factors of 60
The prime factors of a number are all the factors of a number that are also prime.
A prime number can only be divided by one, and itself.
For example, 3 can only be divided by 1 and 3, so it is prime.
1 itself isn’t a prime number. Here’s an article that explains why in a bit more detail.
From the list of factors of 60 above, there are 3 numbers that fit this description:
These 3 numbers are 60’s prime factors.
How to Find the Factors of 60
So, now you know what all the factors of 60 are, but you also need to know how to find them.
Since factors come in factor pairs, once you find one factor, you can easily get another by finding the other half of the factor pair. You can find this other number of the pair by dividing the number you are factorizing by the factor you’ve already found.
For example, once you know that 5 is a factor of 60, you then work out 60÷5=12, so 12 is another factor of 60 making the factor pair (5, 12).
You will have found all the factors when you reach the whole number below the square root of 60.
60’s square root is 7.74… so you only need to check up to 7.
You can check whether a number is a factor using quick and easy tests you can perform in your head to check whether a number is divisible.
Remember, if a number can be divided then you have found a factor!
- All whole numbers are divisible by 1, so there is nothing to check here – 1 is automatically a factor.
- A number is divisibly by 2 if its last digit is a multiple of 2, so either 0, 2, 4, 6, or 8.
- It is divisible by 3 if the sum of its digits is also divisible by 3.
- 4 is a divisor if the last two digits form a number that is also divisible by four. If your number is only two digits long, you can use this trick instead:
A number is divisible by 4 if half of the number is divisible by 2.
- A number is divisible by 5 if its last digit is 5 or 0.
- It can be divided by 6 if it is divisible by both 2 and 3. This is because 6=2×3.
- A number is divisible by 7 if … this one’s a bit tricker that the others:
- Remove the last digit and double it.
- Then subtract the doubled digit from the rest of the number.
- If the result is divisible by 7, then so was the original number.
… and for 60, you don’t need to go any further than this!
You can quickly see that 60 passes the tests for 1, 2, 3, 4, 5, and 6. The only one that might require a bit of work is 7, and it actually fails this one:
And 6 is not divisible by 7, so 60 isn’t divisible by 7.
Now we have found all the factors up to the square root of 60, so you can work out the other factors easily: divide 60 by each of the factors you’ve found to get the other factor in the pair.
- 60 ÷ 1 = 60
- 60 ÷ 2 = 30
- 60 ÷ 3 = 20
- 60 ÷ 4 = 15
- 60 ÷ 5 = 12
- 60 ÷ 6 = 10
That’s it! We’ve found all the factors!
Challenge – Try It Yourself!
These divisibility tests are incredibly powerful when it comes to checking the divisibility of large numbers. For instance…
- It ends in 7, which is an odd number, meaning it’s not a multiple of 2, 4, or 6, and it’s also not a multiple of 5.
- The sum of its digits is 6+3+7+7=23 which is not a multiple of 3, so it’s not a multiple of 3.
- Finally, for divisibility by 7, you get:
And 56=7×8 and so is a multiple of 7, meaning that the original number 6377 is also a multiple of 7.
The only number of 2, 3, 4, 5, 6, and 7 which divides 6377 is 7.
Prime Factorization of 60
The first section established that the prime factors of 60 are 2, 3, and 5.
Whole numbers have a special property: each one can be written as a product of its prime factors in one way and one way only.
This is called its prime factorization. The prime factorization of 60 looks like this:
You can find a number’s prime factorization using a factor tree. Here’s how they work:
- Start by writing down the number you’re factorizing at the top
- Split the number up into two factors, connected to the number above by ‘branches’.
- Don’t choose 1 as a factor – it’s not going to be helpful!
- If any of the factors are prime, circle them.
- Repeat the process for any un-circled number: split numbers up into factor pairs and circle any primes until all the numbers left at the ends of branches have been circled, meaning they’re prime.
- The circled primes that you’re left with are your prime factorization.
This is what you get for 60:
- First you just write down the number 60…
- Then, choose any factor pair of 60, for example (4,15), and split it up. Neither 4 nor 15 are prime, so don’t circle anything yet…
- Split up 4 into 2 and 2 – it’s the only choice. 2 is prime, so get circling!
- Likewise, split up 15 into 3 and 5.
Then, you’re left with
Of course, since these prime factorizations are unique, it shouldn’t matter how you go about doing the factor tree. You could have started with any factor pair of 60 at the first split and you’d end up with the same result.
If you don’t believe it, try it yourself!
Challenge – Try it Yourself!
a) 54 = 2 × 33
b) 130 = 2 × 5 × 13
c) 420 = 22 × 3 × 5 × 7
Here’s what your factor trees might look like.
If you chose different factors at the splitting-up stages, your trees will look different, but you should have the same answer if your math is correct!
Isn’t 60 Interesting
There are 60 seconds in a minute and 60 minutes in an hour: 60 is an important number!
Here are some facts about 60 that you might have seen before:
- The time that it takes for a car to go from 0 to 60 miles per hour is often used to measure the performance of the car. 60mph is also a common speed limit used in lots of countries around the world.
- People who have been married for 60 years celebrate their diamond anniversary. In 2012 Queen Elizabeth II celebrated her diamond jubilee, which marked her 60th year on the throne.
- The highest score that a single dart can score in a game of darts is 60 – by landing on treble 20.
- The length of a bowling lane in ten-pin bowling is exactly 60 feet.
It’s not surprising that the Ancient Babylonians decided to centre their number system around the number 60, since it has some interesting mathematical properties:
To Sum Up (Pun Intended!)
We factorized 60, and found the following 12 factors using divisibility tests:
We also found its prime factorization using a factor tree:
Make sure that you understand both of these methods.
If you’re still not totally sure, have a look at some other examples, like our articles on 18 and 52, which walk you through factorizing some other numbers. You’ll find some interesting facts about those numbers there too!
If you have any solutions to the challenges, questions about the article, or your own fact about the number 60 that we missed here, let us know by leaving a comment below!