Formula
n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖnumber of divisors = (a₁ + 1)(a₂ + 1)…(aₖ + 1)sum of divisors = Π (p^(a+1) − 1) ÷ (p − 1)
- Each pᵢ is a prime and aᵢ is how many times it divides n
Every whole number above 1 is a product of primes in exactly one way, ignoring order.
Worked examples
360
- Whole number (2 or more)
- 360
Prime factorisation of 360
2³ × 3² × 5
Distinct prime factors: 2, 3, 5
Number of divisors: 24
Sum of divisors: 1,170
A prime: 9973
- Whole number (2 or more)
- 9,973
Prime factorisation of 9,973
9,973 (prime)
Distinct prime factors: 9,973
Number of divisors: 2
Sum of divisors: 9,974
How it works
The fundamental theorem of arithmetic says every whole number greater than 1 is a product of primes in exactly one way. 360 = 2³ × 3² × 5, which means 2 divides it three times, 3 twice and 5 once. The calculator divides out the smallest primes in turn, and stops early once what remains is itself prime.
The exponents tell you more than the factors alone. Add 1 to each exponent and multiply to count the divisors — (3 + 1)(2 + 1)(1 + 1) = 24 for 360 — and use the sum formula for the total of all divisors. A number is a perfect square exactly when every exponent is even.
Factorisation underlies the HCF and LCM Calculator and the simplified radical form in the Square Root Calculator. To check a single number quickly use the Prime Number Checker, and to test divisibility by small numbers use the Divisibility Checker.
Assumptions and limits
- Whole numbers from 2 up to 10¹⁵, factorised exactly.
- Numbers that are the product of two very large primes take longer; if a number is too large to handle reliably the tool says so.