How do you find the prime factorization of a number?
Divide the number by the smallest prime that goes in evenly, and keep dividing the quotient until you reach 1. The primes you divided by are the prime factors.
This is called the division ladder, and it's the method the calculator shows. Here it is for 360.
Once 2 stops dividing evenly, move up to 3, then 5, then 7. You never need to try 4 or 6, because any factor of 4 was already removed as a 2.
How do you make a factor tree?
Split the number into any two factors, then keep splitting each branch until every branch ends in a prime. The primes at the ends of the branches are the prime factorization.
For 360 you might start with 36 × 10. Then 36 splits into 6 × 6, and each 6 splits into 2 × 3. The 10 splits into 2 × 5. Reading the ends gives 2, 3, 2, 3, 2, 5, which sorts to 2³ × 3² × 5.
A different first split, like 360 = 8 × 45, leads to the same primes. That's the point of the uniqueness rule: the tree can look different, but the factorization can't.
How do you count the factors of a number?
Add 1 to each exponent in the prime factorization and multiply the results. That gives the total number of factors of a number, including 1 and the number itself.
For 360 = 2³ × 3² × 5¹, the count is (3 + 1) × (2 + 1) × (1 + 1) = 4 × 3 × 2 = 24. The extra 1 is there because each factor can use a prime zero times.
| Number | Prime factorization | Number of factors |
|---|---|---|
| 36 | 2² × 3² | 3 × 3 = 9 |
| 60 | 2² × 3 × 5 | 3 × 2 × 2 = 12 |
| 100 | 2² × 5² | 3 × 3 = 9 |
| 360 | 2³ × 3² × 5 | 4 × 3 × 2 = 24 |
The calculator lists all 24 factors of 360, from 1, 2, 3, 4, 5 and 6 up to 120, 180 and 360. It also shows the 12 factor pairs, such as 15 × 24 and 18 × 20.
A number has an odd count of factors only when it's a perfect square. You can see it in the table: 36 and 100 have 9 factors each, because one factor (6 or 10) pairs with itself.
How can you tell if a number is prime?
Test whether any prime up to the square root of the number divides it. If none does, the number is prime.
You can stop at the square root because factors come in pairs, and one member of each pair is always at most √n. For 97, √97 is a little under 10, so you only test 2, 3, 5 and 7. None of them divides 97, so 97 is prime, and the calculator reports it that way.
What is the prime factorization of 1001?
The prime factorization of 1001 is 7 × 11 × 13. It's a good example of a number that looks prime but isn't.
Testing 2, 3 and 5 fails: 1001 is odd, its digits add to 2, and it doesn't end in 0 or 5. The next prime, 7, works: 1001 ÷ 7 = 143. Then 143 ÷ 11 = 13, and 13 is prime. A related example is 91 = 7 × 13, another number that often gets mistaken for a prime.
What mistakes do people make with prime factorization?
Stopping at a composite number is the most common one. Writing 360 = 2 × 2 × 2 × 45 isn't finished, because 45 is still 3² × 5.
Including 1 as a prime factor is another. The number 1 isn't prime, so it never appears in a prime factorization, and the calculator asks for a number of 2 or more.
Some students also drop a repeated prime. If you write 360 = 2 × 3 × 5, you've listed the distinct primes, but the product is only 30. Multiply your answer back out to catch this.
What is prime factorization used for?
Prime factors are the shortcut behind the GCF, the LCM and simplifying radicals. Once you know the primes, each of those becomes a quick lookup.
For the GCF you take the lowest shared powers, for the LCM the highest powers, and for a square root you pull out pairs. That's why the GCF, LCM and radical calculators on this site all start with a prime factorization step.
Which divisibility rules speed up prime factorization?
A few quick rules tell you whether 2, 3 or 5 divides a number before you do any long division. They cover most of the work in a typical homework problem.
| Prime | Rule | Example |
|---|---|---|
| 2 | The last digit is even | 360 ends in 0 |
| 3 | The digits add to a multiple of 3 | 504: 5 + 0 + 4 = 9 |
| 5 | The last digit is 0 or 5 | 45 ends in 5 |
| 7 | No quick rule, so divide and check | 1001 ÷ 7 = 143 |
Using these on 504: it's even, so divide by 2 three times to reach 63. The digits of 63 add to 9, so divide by 3 twice to reach 7, which is prime. That gives 504 = 2³ × 3² × 7.
How do you use the prime factor calculator?
Enter one whole number of 2 or more, such as 360, and press solve. You'll get the division ladder, the answer in exponent form and the full list of factors.
Exponents appear with a caret, so 2³ × 3² × 5 shows up as 2^3 × 3^2 × 5. If you enter a prime such as 97, the calculator says the number is prime instead of returning a product. The last step multiplies the primes back together, so you can see the factorization gives the original number.