In this lesson, you will learn the factors of 52, what they are, and how to find them.
By the end, you’ll know that factors come in pairs and that every number can be written using a product of prime factors.
You will also be able to find the factors of any number you like, with or without a calculator!
To get started, a factor is a whole number that neatly divides another whole number equal to or bigger than it. The division leaves no remainder.
All Factors of 52
52 has 6 factors:
Every number has 2 fewer proper factors than ‘regular’ factors because the number itself and 1 are crossed off the list.
So as you can expect, 52 has 4 proper factors: 2, 4, 13, and 26.
Factor Pairs of 52
Factor pairs are two factors which can be multiplied together to give the original number.
Most of the time, the factors in a pair are different from each other so most numbers have an even number of factors.
Square numbers are special; their square root is in a factor pair with… their square root! Square numbers always have an odd number of factors because one of their factor pairs has this repeat.
Every number has at least two factors: itself and 1.
52’s factors can be sorted into pairs which multiply to give 52:
What happens if you ignore the convention about including negative numbers?
To get a positive number, you must either multiply two positive or two negative numbers. The negative factors must all be in pairs with negative factors, like a reflection of the pairs above.
Prime Factors of 52
Prime factors are exactly what it says on the tin. They’re factors which are prime!
Prime numbers are only divisible by themselves and 1 so they don’t have any proper factors. Think of primes as a strong foundation that cannot be factorized and broken down.
The prime factorization is where a positive integer is written as a product of its prime factors.
The prime factors of 52 are 2 and 13.
The prime factorization always exists, and no two numbers have the same one.
You could stop here, knowing the factors and prime factorization of 52. But where’s the fun in that?
It’s important to understand how to find factors on your own. Matter of Math won’t be available in your test!
Finding a number’s factors means thinking like a robot. A robot doesn’t just “see” factors, it has to check carefully if each number is a factor until they’ve all been found.
Thinking like a robot is quicker with a robot – a calculator! If you have one, divide the number being factorized by each positive integer, one at a time.
If the division gives a whole number, you’ve found a factor!
Every time you find a factor, find its pair by dividing the original number by the first factor. The dividing number is in a factor pair with the answer on the calculator.
Stop looking for more factors once you reach the original number’s square root.
You can find factors without a calculator too, using the same idea. The divisions might take longer by hand, but divisibility rules can help – especially if the number being factorized is super big!
1: All integers, or whole numbers, are divisible by 1
2: All even integers are divisible by 2
3: Add up the digits. If the sum is divisible by 3, the original number is also
4: Just look at the last two digits. If this number is divisible by 4, the original number is too
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: Split the last digit from the number to get a shorter number and a single digit. Calculate shorter number minus 2×(last digit). If 7 divides this result, the original number is also divisible by 7
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
How to Find the Factors of 52
Ready to think like a robot? Work along with this example to make sure you understand the divisibility rules and method.
1: always a factor
1’s pair is (1, 52)
2: a factor because 52 is even
2’s pair is (2, 26)
3: not a factor because 5+2=7 which is not divisible by 3
4: the divisibility rule doesn’t simplify things, but 52÷4=13 so 4 is a factor
4’s pair is (4, 13)
5: not a factor because 52’s last digit is not 0 or 5
6: not a factor because 3 is not a factor
7: not a factor because 5-2×2=1 which is not divisible by 7
Stop here because eight is bigger than the square root of 52, so all the factors have been found.
This has given 6 factors: 1, 2, 4, 13, 26, and 52. Just what you needed to find!
It’s a balancing act in the factor pairs! The square root is like the pivot in a seesaw.
If the first factor is smaller than the square root, the second factor must be bigger than the square root, or their product will be too small.
If the first factor is bigger than the square root, the second factor must be smaller than the square root, or their product will be too big.
Beyond the square root, any factors found are already in pairs with factors smaller than the square root.
Prime Factorization of 52
Every number can be written as a multiplication of its prime factors.
The method for prime factorization is very similar to what you’ve just seen.
This time, however, you don’t need to find all the factors; one proper factor pair is enough at each stage. Phew!
Start by picking your favorite of the number’s proper factor pairs. If the factors are both prime, you can stop!
If either or both aren’t prime, find their proper factor pair. If these are both prime, then you can stop!
Repeat this process until you have a list of only prime factors. It might help to imagine the original number as a tree, with its factors splitting off like branches.
The prime factorization is the product of these prime factors. If the same factor is there more than once, don’t skip it!
One more thing: it doesn’t matter which proper factor pairs you choose to start, the prime factorization will always be the same – it isn’t fussy about which route you take to find it!
You can use the result from the factor tree to write the prime factorization.
Start with the number you’re working with – 52.
Then choose two factors that multiply together to make 52. It doesn’t matter which two you choose, let’s go with (2, 26).
2 is a prime factor so we don’t need to go any further with that one.
Next, choose a factor pair of 26. Let’s use (2, 13)
2 and 13 are both prime factors.
If we gather all the prime factors:
And finally, tidy this up using exponent notation:
Here are two related problems for you to work through, to make sure you not only understand the method, but can use it yourself.
1: always a factor
1’s pair is (1, 96)
2: a factor because 96 is even
2’s pair is (2, 48)
3: a factor because 9+6=15 which is divisible by 3
3’s pair is (3, 32)
4: a factor because 96/4=24
4’s pair is (4, 24)
5: not a factor because 96’s last digit is not 0 or 5
6: a factor because 96 is divisible by both 2 and 3
6’s pair is (6, 16)
7: not a factor because 9-2×6=-3 which is not divisible by 7
8: a factor because 96/4=24 and 24 is even, so it is divisible by 2
8’s pair is (8, 12)
9: not a factor because 9+6=15 which is not divisible by 9
96’s factors are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.
The prime decomposition, or factorization, of 96 is:
Isn’t 52 Interesting?
- As mentioned above, there are 52 weeks in a year. 7 isn’t actually a factor of 365 or 366 so there are 1 or 2 days leftover!
- Sticking with the calendar theme: the Mayan calendar uses cycles of 52 standard years. We’re all very young in Mayan ‘years’!
- A standard deck of playing cards has 52 cards. A good practical joke is to ask someone if they want to play ’52 card pickup’ with you, then throw the deck on the floor so they have to put them all away!
To Sum Up (Pun Intended!)
We hope you feel a little smarter now you’ve finished learning about factors! The first steps to becoming a factor-master were the definitions.
Factors are positive integers (whole numbers) that perfectly split an equal or larger number, leaving no remainder. They can be arranged into factor pairs whose products are the original number.
Proper factors are factors that aren’t 1 or the original number – these are the interesting ones!
Another easy definition is prime factors. These are factors that are also prime! The combination of prime factors which multiply to make the original number is the prime factorization.
You had a peek ahead at the factors of 52 and its prime factorization. Then, you learned the ‘robot technique’ to find them for yourself.
Keep practicing and make use of the resources available here to build your confidence. Got a question or fun 52-related fact? Leave a comment below.