This lesson will teach you what the factors of 26 are, how to find them, and the different ways we can use them.
A factor is a number that can divide another number exactly without leaving a remainder.
Once you’ve learned how to find the factors of a simple number like 26, you’ll be able to find the factors of any number you like!
All Factors of 26
The factors of 26 are:
This means that dividing 26 by 1, 2, 13, or 26 will give a whole number answer. Other numbers are not factors because they don’t exactly divide 26; they leave remainders.
Factor Pairs of 26
Factor pairs are combinations of factors that multiply together to make the original number.
Factors are picky; they are only in a pair with one other factor. Some factors, as you will see below, are in pairs with themselves!
26 has 4 factors which give 2 pairs:
26 = 2 × 13
So the factor pairs for 26 are:
What happens if we bend the factor definition to include negative integers? The number of factors doubles!
Multiplying negative numbers still gives the same positive number because:
negative × negative = positive
Including negative factors, 26 can be written as the following pair products:
Prime Factors of 26
Prime numbers always have two factors because they are only divisible by themselves and 1.
Prime factors, as you’ve probably worked out, are all the factors that are prime numbers.
The prime factors of 26 are:
Even though you’ve now seen the answers, you’re far from done. Math is all about being able to use the process.
The easiest way to find a number’s factors is to use a calculator. Checking if each number is a factor can be a slow process sometimes but there are a few tricks to speed you along.
Start by writing down 1, which always divides a whole number.
Then divide the target number by each number in increasing order, counting up from 2, so you don’t miss any out.
Numbers can surprise you by having more factors than expected.
Another trick is to stop checking once you reach the square root of the original number. The factors bigger than the square root are in pairs with numbers smaller than the square root.
You might not always have a calculator – I’m looking at you, non-calculator exams! – but there are some rules to speed up dividing numbers by hand.
These are the divisibility rules to check if a number is divisible by the integers between 1 and 10. They are particularly helpful when dealing with big numbers.
1: All integers are divisible by 1.
2: All even integers are divisible by 2.
3: Add up the digits. If the sum is divisible by 3, so is the original number.
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: Take off the last digit, double it, and take it away from the remaining number. If the result can be divided by 7, so can the original number. Repeat this process until you have a small enough number to check easily.
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 number.
10: If the last digit is 0, the number is divisible by 10.
Open the image below to get and print a free set of divisibility rules up to 15.
How to Find the Factors of 26
Now you’ve seen the method, let’s apply it to 26!
The square root of 25 is 5, and the square root of 26 is 6, so the square root of 26 is somewhere in between 5 and 6.
This means we should check whether 26 is divisible by all the numbers up to 5.
1: all numbers are divisible by 1 so it is a factor.
1’s pair is (1, 26)
2: 26 is even so 2 is a factor.
2’s pair is (2, 13)
3: 2 + 6 = 8 and 8 is not divisible by 3 so 3 is not a factor
4: it’s not a factor. You can see that it isn’t by dividing 26 by 2, then dividing by 2 again. The result is a decimal so 26 is not divisible by 4.
5: it’s not a factor because 26’s last digit is neither 5 nor 0.
We have found all the factors: 1, 2, 13, 26.
Prime Factorization of 26
Prime factorization is a bit trickier. The idea is the same: numbers can be broken down into factor pairs using a calculator or divisibility rules – but there are often more steps.
The first step is to factorize the number and pick a proper factor pair. This choice doesn’t matter; the result will always be the same!
The second step is to check if these factors are prime. Are they? Great, you’re done.
If either factor is not prime, find the factor’s factors and repeat the process.
Prime factorization breaks numbers down until all that’s left is a product of the simplest possible building blocks.
Your answer should not include 1 because it doesn’t add any information.
However, repeats of the same prime factor must be included or the product won’t make the original number. You can tidy up repeated factors by expressing them as exponents.
Prime numbers don’t have a prime factorization. They are perfect as they are!
26 has proper factor pair (2, 13).
Both 2 and 13 are prime numbers, so neither can be factorized further.
Both factors are prime, so we are done! That was easy.
The prime factorization of 26 is
Let’s get stuck into something more challenging, using exactly the same method, to solidify what you have learned so far.
1: always a factor.
1’s pair is (1, 176)
2: 176 is even so 2 is a factor
2’s pair is (2, 88)
3: 1+7+6= 14 which isn’t divisible by 3 so 3 isn’t a factor
4: 76 is divisible by 4 so 176 is.
4’s pair is (4, 44)
5: 176’s last digit is 6 so it isn’t divisible by 5
6: not a factor because 3 isn’t a factor
7: split 176 into 17 and 6. Doubling 6 and subtracting the result from 17 gives 5 which is not divisible by 7, so 176 isn’t
8: dividing 176 by 2 gives 88. 88 is divisible by 4 so 176 is divisible by 8
8’s pair is (8, 22)
9: 1+7+6=14 which isn’t divisible by 9 so 9 is not a factor of 176
10: 176’s last digit is not 0 so 10 is not a factor
11: is a factor because 176/11 gives an integer
11’s pair is (11, 16)
12: not a factor because 176 is not divisible by both 3 and 4
13: not a factor because 176/13 doesn’t give an integer
We stop here because the square root of 176 is less than 14.
The factors of 176 are: 1, 2, 4, 8, 11, 16, 22, 44, 88, 176.
Let us know how you got on in the comments at the bottom of this lesson.
Isn’t 26 Interesting?
Numbers are all around us in day-to-day life and 26 is no exception. You might have noticed the following fun facts:
- There are 26 letters in the Latin alphabet – that’s the alphabet you’re reading in now! – but other alphabets are different. The Greek alphabet has 24 letters, the Russian has 32 and there are only a teeny tiny 54,648 Chinese characters!
- A marathon is just over 26 miles long. The length of this exhausting race varied between about 24 and 26 miles at different competitions, until 1921 when 26.2 miles became the official distance. Your legs would be feeling every yard!
- The 26th element in the periodic table is iron. Iron is important for our health because it’s in our red blood cells, helping transport oxygen from the lungs to all the cells in our bodies.
To Sum Up (Pun Intended!)
I hope you understood and enjoyed learning about all the different kinds of factors!
You learned the definition of factors, factor pairs, and prime factors. You then saw the factors of 26, its prime factorization, and learned a killer technique to factorize any number.
A calculator is the quickest way to find all a number’s factors but failing that, you’ll need divisibility rules and a little patience!
Factors always come in pairs. The square root of a square number is in a factor pair with itself.
This is useful because we can stop searching for factors once we reach our number’s square root. The bigger factors have already been found, in pairs with smaller factors.
Prime factorization is an extension of regular factoring. The number is repeatedly factorized until it’s in its simplest form.
If you’ve got any questions, or just want to show off a huge number you’ve factorized, please leave a comment below!