To find out the factors of 48, you need to know how to factorize. A factor is a number that goes into another number without leaving a remainder.
For example, 2 is a factor of 4 because it goes into 4 twice!
In this lesson, we will tell you all the factors as well as give you a simple method so that you can find out the factors of any number you want.
We will also look at factors coming in pairs, primes, and writing 48 as simply as possible using prime factorization.
All Factors of 48
The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48!
What are the Factor Pairs of 48?
A factor pair of 48 is a pair of whole numbers that multiply together to give 48. Pairs are usually written in parentheses.
Here is a list of 48’s factor pairs:
For every factor pair of positive numbers, there is also a pair of negative numbers, because when you multiply two negative numbers, you always get a positive number!
From the factor pairs above you can get the following list of positive whole number that 48 is divisible by:
From now on, a factor of 48 refers to one of the ten numbers in the list here.
You might see it called a divisor, which is just another word for a factor.
Prime Factors of 48
A prime factor is a positive whole number that is both prime and a factor of the original number.
A number is prime if it has exactly two factors: 1 and itself.
From this definition, and looking at the list above, you can see that for 48, there are only two prime factors:
How to Find the Factors of 48
There are several methods you can use to find the factors of a number. Here, we’ll show a method using divisibility tests.
Divisibility tests are simple mental math tricks that tell you if a number is divisible by another.
Once one factor has been found using a divisibility test, you can find the second because factors come in pairs.
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!
To make sure no numbers are missed, you must check for divisibility by every whole number up to the square root of 48, which is 6.928…<7, so check up to 6.
1: All numbers are divisible by 1, so no test is needed!
This gives the most straightforward factor pair that all numbers have: 1 and themselves, (1,48)
2: If the last digit is one of 0, 2, 4, 6, 8.
This applies to 48, meaning that 48 is divisible by 2.
You can fine the other factor in the pair by finding 48÷2=24 so the pair is (2,24).
3: If adding up all the digits gives a multiple of 3.
This test might seem unhelpful at first, but it can drastically reduce the size of the numbers you are dealing with. Then, you can quickly see whether or not it is a multiple of 3 you recognize.
= 3 × 4
The digit sum of 48 is 12, which is divisible by 3, so 48 is also divisible by 3. 48÷3=16, and so (3,16) is the pair.
4: If both the number and its half are divisible by 2.
We’ve just seen that 48 is divisible by 2, and half of 48 is 24. The last digit of 24 is 4, so 24 is also divisible by 2, meaning 48 is divisible by 4.
After calculating 48÷4=12, you can get the next pair: (4,12).
There is another test for divisibility by 4: look at the last two digits for larger numbers. If they form a number that’s divisible by 4, then the entire number’s divisible by 4.
5: If the last digit is either 0 or 5.
This is probably the easiest of all the divisibility tests, and 48 clearly fails, so 48 is not divisible by 5.
6: If it’s both divisible by 2 and 3.
Since 6=2×3 any number that’s got a factor of 6 must also have a factor of 2 and a factor of 3, and vice versa.
Having already completed the tests on 48 for divisibility by 2 and 3, both of which came back ‘positive,’ 48 must be divisible by 6. 48÷6=8, and so our last pair is (6,8).
This method has produced the same list of positive factor pairs shown at the beginning:
These divisibility tests are powerful, particularly when it comes to larger numbers. Have a look at the number 46,620.
2) The last digit is 0, so its divisible by 2, 5, and 10.
3) The digits sum is 4+6+6+2+0=18=3×6, a multiple of 3, so it’s divisible by 3.
4), 5) The last two digits are 20=4×5, so it’s divisible by both 4 and 5.
6) 46,620 is divisible by 2 and 3, so it is divisible by 6.
So, 46,620 is divisible by 2, 3, 4, 5, and 6.
The Prime Factorization of 48
Remember that there are 2 prime factors of 48:
Every number can be written as a product of its prime factors in a unique way. Mathematicians call this the fundamental theorem of arithmetic.
The resulting combination of prime factors is called the prime factorization, or sometimes prime decomposition.
1. Start by writing the number you want to factorize.
2. Draw two branches coming down from the number, writing numbers that make up a factor pair at the end.
3. If either of the factors is prime, circle it.
4. For each of the remaining numbers in the tree that are not circled, repeat the process from step 2 by splitting up the numbers.
5. Once there’s a circled number at the end of each branch, you’re done. The circled prime numbers are the prime factorization.
Because prime decompositions are unique, it doesn’t matter which pairs you choose to split up numbers during the process – you’ll always end up with the same factorization.
Here’s a factor tree for 48.
By looking at the circled prime factors, you can show that the prime factorization of 48 is:
To simplify this you can write:
You can use the prime factorization of a number to work out how many factors it has in total without having to work out what they all are!
Each factor is made up of some of the prime factors multiplied together – counting the number of factors is the same as counting the number of different combinations of prime factors from the prime decomposition.
For 48, each factor can have between 0 and 4 copies of the prime factor 2, and either 0 or 1 copy of the prime factor 3.
For example, 8, which divides 48 into 6, has the prime decomposition 23. This means it has 3 copies of 2 and zero copies of 3.
Also, having zero of both prime factors gives 1 because any number to the power 0 is 1.
So, there are 5 choices for the number of 2’s in a factor of 48, and 2 options for the number of 3’s, resulting in 5×2=10 factors in total.
Have a go at this yourself.
a) 90=2×32×5 so 2 choices for no. of 2’s, 3 choices for no. of 3’s, and 2 choices for no. of 5’s. In total, 2×3×2=12 factors.
b) 120=23×3×5, so with 4 choices for 2’s, and 2 choices each for 3’s and 5’s. In total, 4×2×2=16 factors.
c) 320=24×5, so there are 7 choices for the no. of 2’s, and 2 choices for the no. of 5’s. In total, 7×2=14 factors.
You’ll find this method easy when finding how many factors a huge number has.
If a number is square, all the prime factorization exponents are even, meaning there is an odd number of choices for the number of each prime in a factor.
The total number of factors is the product of each of these odd numbers of choices, which will itself be an odd number since odd times odd is odd.
Isn’t 48 Interesting?
Here are some interesting facts about the number 48 from outside the world of math:
- 48 hours are a significant length of time, since it is the number of hours in two 24 hour day/night periods.
- The playing time for an NBA game is 48 minutes, which does not include time spent for timeouts, fouls, substitutions, etc.
… and a mathematical fact about 48, completely unrelated to factors…
- 48 is the smallest number with 10 different factors. This makes it a highly composite number , which are numbers that have more factors than any number smaller than them.
To Sum Up (Pun Intended!)
We found all the factors of 48 using some simple rules called divisibility tests that can tell you quickly when a large number is divisible by a particular, small number.
The result was the following list of 10 factors:
Then, to find the unique prime factorization, we used the method of factor trees, and found:
Make sure that you understand how each of these methods works. Test yourself – choose a number, try and find all its factors, and then its prime factorization.
If you have any questions about the material covered here, or if you have a fact about the number 48 that we’ve missed, please let us know in the comments below!