Prime Factorization Calculator
- Updated
- Formula shown
- Free, no sign-up
Results
The prime factorization of 360 is 2³ × 3² × 5
Divide by the smallest prime that goes in exactly, and repeat until you reach 1: 2 | 360 2 | 180 2 | 90 3 | 45 3 | 15 5 | 5 | 1 Read the primes down the left side: 360 = 2 × 2 × 2 × 3 × 3 × 5 With exponents: 360 = 2³ × 3² × 5
How to use the prime factorization calculator
- Enter a whole number from 2 up to 18 digits.
- Choose how to show the working: a division ladder (divide by the smallest prime again and again) or a factor tree (split into two factors until every branch is prime).
- Read the factorization with exponents, the expanded product, and whether the number is prime. Copy the working with one click.
Prime factorization formula
A prime is a whole number greater than 1 whose only factors are 1 and itself: 2, 3, 5, 7, 11 and so on. The fundamental theorem of arithmetic says every whole number n greater than 1 can be written as a product of primes p₁, p₂, … raised to whole-number exponents a, b, …, and that this product is unique apart from the order of the factors. That is why the prime factorization works as a fingerprint of a number and is the starting point for factors, the GCF, the LCM and simplifying square roots.
Worked example: 360
Division ladder. Divide by 2 while you can, then by 3, then by 5, writing each quotient underneath:
- 360 ÷ 2 = 180
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
Factor tree. The calculator splits each number into two factors as close in size as possible:
360
├── 18
│ ├── 3 (prime)
│ └── 6
│ ├── 2 (prime)
│ └── 3 (prime)
└── 20
├── 4
│ ├── 2 (prime)
│ └── 2 (prime)
└── 5 (prime) Both methods use three 2s, two 3s and one 5, so 360 = 2³ × 3² × 5. Any other tree, such as 360 = 10 × 36, ends with the same primes.
Prime numbers below 100
There are 25 primes below 100, listed here by tens.
| Range | Primes |
|---|---|
| 0–9 | 2, 3, 5, 7 |
| 10–19 | 11, 13, 17, 19 |
| 20–29 | 23, 29 |
| 30–39 | 31, 37 |
| 40–49 | 41, 43, 47 |
| 50–59 | 53, 59 |
| 60–69 | 61, 67 |
| 70–79 | 71, 73, 79 |
| 80–89 | 83, 89 |
| 90–99 | 97 |
Prime factorizations of common numbers
| Number | Prime factorization |
|---|---|
| 12 | 2² × 3 |
| 18 | 2 × 3² |
| 24 | 2³ × 3 |
| 36 | 2² × 3² |
| 48 | 2⁴ × 3 |
| 60 | 2² × 3 × 5 |
| 72 | 2³ × 3² |
| 84 | 2² × 3 × 7 |
| 90 | 2 × 3² × 5 |
| 96 | 2⁵ × 3 |
| 100 | 2² × 5² |
| 120 | 2³ × 3 × 5 |
| 144 | 2⁴ × 3² |
| 180 | 2² × 3² × 5 |
| 210 | 2 × 3 × 5 × 7 |
| 256 | 2⁸ |
| 500 | 2² × 5³ |
| 1000 | 2³ × 5³ |
| 1001 | 7 × 11 × 13 |
| 2024 | 2³ × 11 × 23 |
Numbers that look prime but aren't
Odd numbers that don't end in 5 are easy to mistake for primes. Each of these has a prime factor no larger than its square root:
- 51 = 3 × 17
- 57 = 3 × 19
- 87 = 3 × 29
- 91 = 7 × 13
- 119 = 7 × 17
- 133 = 7 × 19
- 143 = 11 × 13
- 221 = 13 × 17
Tips for factoring by hand
- Only test primes up to the square root. If no prime up to √n divides n, then n is prime. For 97, testing 2, 3, 5 and 7 is enough, because 11² = 121 is already larger than 97.
- Use divisibility rules first. Even numbers divide by 2, numbers whose digits add up to a multiple of 3 divide by 3, and numbers ending in 0 or 5 divide by 5.
- Check your answer by multiplying back. 2³ × 3² × 5 = 8 × 9 × 5 = 360.
- Prime factors make other problems quick: the LCM calculator and GCF calculator compare the exponents, the factor calculator builds every factor from them, and the root calculator uses them to simplify square roots.
Frequently asked questions
What is the prime factorization of 360?
360 = 2³ × 3² × 5, which is 2 × 2 × 2 × 3 × 3 × 5. Dividing by the smallest prime each time gives 360 → 180 → 90 → 45 → 15 → 5 → 1, using the divisors 2, 2, 2, 3, 3 and 5.
Is 1 a prime number?
No. A prime has exactly two different factors, 1 and itself, and 1 has only one. Leaving 1 out also keeps prime factorizations unique: if 1 were prime you could write 6 = 2 × 3 = 1 × 2 × 3 = 1 × 1 × 2 × 3, and so on.
Is 91 prime?
No. 91 looks prime but equals 7 × 13. To test a number by hand, try dividing by each prime up to its square root; for 91 that means 2, 3, 5 and 7, and 7 goes in exactly.
Does every number have only one prime factorization?
Yes. The fundamental theorem of arithmetic says every whole number greater than 1 is a product of primes in exactly one way, apart from the order. A factor tree may split 360 as 18 × 20 or as 10 × 36, but both end with three 2s, two 3s and one 5.
How does the calculator factor large numbers?
It divides out every prime below 1,000, checks what is left with the Miller–Rabin test (exact for numbers of up to 18 digits with the bases used), and splits any remaining composite with Pollard's rho method. Even a product of two 9-digit primes, such as 999,999,929 × 999,999,937, is factored in a fraction of a second.