To find the prime factorization of a number, divide it by the smallest prime that fits, then divide the result again, and keep going until you reach 1. The primes you divided by are the answer. For 360 that's 2³ × 3² × 5. Below you'll find the step-by-step method, divisibility shortcuts, a table of common factorizations, and how the same factors give you the GCF and LCM.
A prime number has exactly two divisors: 1 and itself. The first few are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29, and there are 25 of them below 100. Every whole number above 1 that isn't prime is composite, which means it can be broken into smaller primes multiplied together. That breakdown is its prime factorization.
There's only ever one answer. Split 60 as 6 × 10 or 4 × 15 and keep breaking the pieces down, and you land on 2 × 2 × 3 × 5 either way. That's the fundamental theorem of arithmetic, and it means any method you like will work, including the factor trees most schools teach first.
The most reliable way by hand is repeated division. Start with the smallest prime, 2, and divide as many times as it goes in evenly. Then move to 3, then 5, then 7, and keep going until what's left is 1.
Here's 360 worked through:
Collect the divisors: 2 × 2 × 2 × 3 × 3 × 5, or 2³ × 3² × 5 in exponent form. Multiply them back together and you should get 360.
There's a shortcut that saves a lot of time on bigger numbers. You only need to test primes up to the square root of whatever's left. Take 221. Its square root is about 14.9, so you only try 2, 3, 5, 7, 11 and 13. Thirteen works (221 = 13 × 17), and if nothing up to 13 had worked, you'd know 221 was prime without testing anything else.
You don't have to do long division to find out whether a small prime fits. These quick checks cover the primes you'll test most:
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 3,584 is divisible by 2 |
| 3 | Digits add up to a multiple of 3 | 471: 4 + 7 + 1 = 12, so yes |
| 5 | Last digit is 0 or 5 | 1,235 is divisible by 5 |
| 7 | Double the last digit, subtract it from the rest; result divisible by 7 | 343: 34 − 6 = 28, so yes |
| 11 | Alternating sum of digits is a multiple of 11 (including 0) | 1,331: 1 − 3 + 3 − 1 = 0, so yes |
| 13 | Add 4 × the last digit to the rest; result divisible by 13 | 221: 22 + 4 = 26, so yes |
Here are prime factorizations for numbers that show up constantly in fractions, measurements and time. The last column counts every divisor, not just the primes. You can get that count straight from the exponents: add 1 to each exponent and multiply. For 360 = 2³ × 3² × 5¹, that's 4 × 3 × 2 = 24 divisors.
| Number | Prime factorization | Number of divisors |
|---|---|---|
| 24 | 2³ × 3 | 8 |
| 36 | 2² × 3² | 9 |
| 48 | 2⁴ × 3 | 10 |
| 60 | 2² × 3 × 5 | 12 |
| 72 | 2³ × 3² | 12 |
| 84 | 2² × 3 × 7 | 12 |
| 90 | 2 × 3² × 5 | 12 |
| 100 | 2² × 5² | 9 |
| 120 | 2³ × 3 × 5 | 16 |
| 144 | 2⁴ × 3² | 15 |
| 180 | 2² × 3² × 5 | 18 |
| 360 | 2³ × 3² × 5 | 24 |
| 1,001 | 7 × 11 × 13 | 8 |
| 1,024 | 2¹⁰ | 11 |
Notice how many divisors 60 and 360 have for their size. That's a big reason we still use 60 minutes in an hour and 360 degrees in a circle: both split evenly into halves, thirds, quarters, fifths and sixths. Perfect squares like 36 and 100 are the only numbers with an odd divisor count.
This is where factoring pays off. Once you've written two numbers as primes, the greatest common factor (GCF, also called GCD) and the least common multiple (LCM) fall out almost for free:
Take 48 and 180. 48 = 2⁴ × 3 and 180 = 2² × 3² × 5. They share 2 and 3, so the GCF is 2² × 3 = 12. The LCM uses 2⁴, 3² and 5, which gives 720. A quick check: GCF × LCM always equals the two numbers multiplied together, and 12 × 720 = 8,640 = 48 × 180.
| Pair | Factorizations | GCF | LCM |
|---|---|---|---|
| 12 and 18 | 2² × 3 and 2 × 3² | 6 | 36 |
| 24 and 36 | 2³ × 3 and 2² × 3² | 12 | 72 |
| 60 and 72 | 2² × 3 × 5 and 2³ × 3² | 12 | 360 |
| 84 and 126 | 2² × 3 × 7 and 2 × 3² × 7 | 42 | 252 |
| 48 and 180 | 2⁴ × 3 and 2² × 3² × 5 | 12 | 720 |
| 15 and 28 | 3 × 5 and 2² × 7 | 1 | 420 |
When the GCF is 1, as with 15 and 28, the numbers are called coprime. They share no primes at all, so the LCM is just their product.
The everyday answer is fractions. To reduce 84/126, factor both parts, find the GCF (42), and divide through to get 2/3. To add 7/12 and 5/18 you need a common denominator, and the LCM of 12 and 18 (36) is the smallest one that works. The fraction calculator does both steps for you, but knowing where the numbers come from helps you spot a wrong answer.
Ratios work the same way. A ratio of 48:180 simplifies to 4:15 once you divide both sides by 12, and the ratio calculator handles that reduction automatically.
Enter a whole number and get its prime factorization, every divisor, and whether it's prime.
Factoring Calculator →Divide by 2 until you can't, then 3, then 5, and keep climbing through the primes until you reach 1. Only test up to the square root of what's left. Write the result with exponents, and you've got everything you need for GCFs, LCMs, reducing fractions and counting divisors.
Divide the number by the smallest prime that goes in evenly, starting with 2, and repeat with the result. Move up to 3, 5, 7 and so on until you reach 1. The primes you divided by are the prime factorization. For 360, that's 2 × 2 × 2 × 3 × 3 × 5, or 2³ × 3² × 5.
72 = 2 × 2 × 2 × 3 × 3, written as 2³ × 3². It has 12 divisors in total.
No. A prime has exactly two different divisors, and 1 has only one. Leaving 1 out is also what keeps every number's prime factorization unique.
Factor both numbers, then multiply the primes they share, using the smaller exponent for each. For 48 = 2⁴ × 3 and 180 = 2² × 3² × 5, the GCF is 2² × 3 = 12.
Stop once the prime you're testing is bigger than the square root of the number that's left. If nothing up to that point divides evenly, the remaining number is prime and it's the last factor.