worldofcalcsPrime Factorization Calculator
Division steps
| Divide by | Quotient |
|---|---|
| 360 | |
| 2 | 180 |
| 2 | 90 |
| 2 | 45 |
| 3 | 15 |
| 3 | 5 |
| 5 | 1 |
All divisors
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
What is prime factorization?
Prime factorization writes a number as a product of prime numbers, the numbers greater than 1 whose only factors are 1 and themselves (2, 3, 5, 7, 11 and so on). Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors. It's used to find the HCF and LCM, simplify fractions and square roots, and count a number's divisors.
Division method and divisor formulas
Divide by the smallest prime that goes in exactly, and keep dividing the quotient until you reach 1. The primes you divided by are the prime factors. Each divisor picks every prime 0 up to k times, which gives the divisor count formula.
Tips and shortcuts
- Divisibility rules speed things up: a number is divisible by 2 if it ends in an even digit, by 3 if its digits add up to a multiple of 3, and by 5 if it ends in 0 or 5.
- You only need to try primes up to the square root of the number. If none divides it, the number is prime.
- A factor tree gives the same answer as the division method. Split the number into any two factors and keep splitting until every branch ends in a prime.
- The HCF of two numbers multiplies the primes they share, at the lower power. The LCM multiplies every prime at the higher power.
- A number is a perfect square when every power in its prime factorization is even, such as 144 = 2⁴ × 3².
Worked example: prime factors of 360
- The smallest prime that divides 360 is 2: 360 ÷ 2 = 180
- Keep dividing by 2: 180 ÷ 2 = 90, then 90 ÷ 2 = 45
- 45 is odd, so move to 3: 45 ÷ 3 = 15, then 15 ÷ 3 = 5
- 5 is prime: 5 ÷ 5 = 1, so stop
- 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5
- Number of divisors: (3 + 1)(2 + 1)(1 + 1) = 24
Prime factorizations of common numbers
Numbers that come up often in school exercises, with their divisor counts.
| Number | Prime factorization | Divisors |
|---|---|---|
| 12 | 2² × 3 | 6 |
| 24 | 2³ × 3 | 8 |
| 36 | 2² × 3² | 9 |
| 48 | 2⁴ × 3 | 10 |
| 60 | 2² × 3 × 5 | 12 |
| 72 | 2³ × 3² | 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 has no prime factors. A prime number has exactly 2 divisors.