Factorization
2^3 × 3^2 × 5
Divisors
24
Prime
No
Works best with numbers up to 10 billion. Larger numbers may take a moment.
360 = 2^3 × 3^2 × 5
3
24
1,170
No
= 8
= 9
n = p1^a1 × p2^a2 × ... × pk^akEvery integer n > 1 can be written uniquely as a product of prime powers, where p1 < p2 < ... < pk are primes and each ai >= 1.
Where:
n= The number to factorizepi= Distinct prime factorsai= Exponent (how many times pi divides n)d(n) = (a1 + 1)(a2 + 1)...(ak + 1)The total count of positive divisors of n, derived from its prime factorization.
Where:
d(n)= Total number of divisors of nai= Exponent of the i-th prime factorσ(n) = Π (pi^(ai+1) - 1) / (pi - 1)The sum of all positive divisors of n, computed from its prime factorization.
Where:
σ(n)= Sum of all divisors of npi= i-th prime factorai= Exponent of pi in the factorizationInputs
Result
360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5. Result: 2³ × 3² × 5. Divisors: (3+1)(2+1)(1+1) = 24.
Inputs
Result
84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 ÷ 3 = 7. Result: 2² × 3 × 7. Divisors: (2+1)(1+1)(1+1) = 12.
Inputs
Result
97 is not divisible by 2, 3, 5, or 7 (7² = 49 < 97 < 100 = 10²). The next prime to test would be 11, but 11² = 121 > 97, so 97 is prime.
Prime factorization is expressing a number as a product of prime numbers. Every integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic). For example, 360 = 2^3 × 3^2 × 5.
| Number | Prime Factorization | Unique Primes |
|---|---|---|
| 12 | 2² × 3 | 2 |
| 60 | 2² × 3 × 5 | 3 |
| 360 | 2³ × 3² × 5 | 3 |
| 1000 | 2³ × 5³ | 2 |
Use trial division: start by dividing by the smallest prime (2), then 3, 5, 7, and so on. Keep dividing by the same prime until it no longer divides evenly, then move to the next. Stop when the quotient is 1.
| Step | Division | Result |
|---|---|---|
| 1 | 360 ÷ 2 | 180 |
| 2 | 180 ÷ 2 | 90 |
| 3 | 90 ÷ 2 | 45 |
| 4 | 45 ÷ 3 | 15 |
| 5 | 15 ÷ 3 | 5 (prime, done) |
If n = p1^a1 × p2^a2 × ... then the total number of divisors is (a1+1)(a2+1).... Each divisor is formed by choosing an exponent from 0 to ai for each prime pi. For 360 = 2^3 × 3^2 × 5, there are (3+1)(2+1)(1+1) = 24 divisors.
| Number | Factorization | Divisor Count Formula | Count |
|---|---|---|---|
| 12 | 2² × 3 | (2+1)(1+1) | 6 |
| 360 | 2³ × 3² × 5 | (3+1)(2+1)(1+1) | 24 |
| 1000 | 2³ × 5³ | (3+1)(3+1) | 16 |
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The number 2 is the only even prime.
| Number | Prime? | Reason |
|---|---|---|
| 2 | Yes | Only divisors are 1 and 2 |
| 15 | No | 15 = 3 × 5 |
| 17 | Yes | Not divisible by 2, 3, or 4 (4² > 17) |
| 91 | No | 91 = 7 × 13 |
Last Updated: Mar 9, 2026
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