What is a Divisors & Multiples Calculator?
This calculator finds all divisors of a number, checks whether it's prime, and lists its multiples.
How it's calculated
A divisor is a number that divides evenly; a multiple is an integer multiple of a given number. A number is prime if its only divisors are 1 and itself.
Step-by-step example
How many divisors does 36 have?
- Step 1 Prime factorize: 36 = 2²×3²
- Step 2 Add 1 to each exponent and multiply: (2+1)×(2+1)
- Step 3 Result: 9 divisors (1,2,3,4,6,9,12,18,36)
Interpretation: Prime factorization gives the divisor count instantly, without checking each number individually.
Why do prime numbers matter so much?
Primes are the building blocks that construct every whole number. Every integer greater than 1 can be broken down into a product of primes in exactly one way (the Fundamental Theorem of Arithmetic) — 12 = 2×2×3, and there's no other combination. Primes are, in a very real sense, the atoms of numbers — and as seen in factoring, this exact property underlies modern cryptography.
Good to know
- There are infinitely many primes — proven by Euclid over 2,300 years ago, using an elegantly simple argument: assume a finite list of all primes exists, multiply them all together and add 1, and you get a new number divisible by none of them, a contradiction.
- Despite that, nobody knows the exact pattern of where primes occur — the Riemann Hypothesis, one of math's greatest unsolved problems, deals with this question and carries a million-dollar prize for a proof.
- 1 is not considered prime — if it were, factorizations like 12 = 2×2×3 = 1×2×2×3 would no longer be unique, breaking the whole system.
- A perfect number equals the sum of its divisors excluding itself — 6 (1+2+3) and 28 (1+2+4+7+14) are examples. Whether any odd perfect number exists remains unsolved.
- The largest known primes run into tens of millions of digits, discovered through a worldwide distributed computing project (GIMPS) that anyone can contribute spare computer time to.
Frequently asked questions
Q. Is 1 a prime number?
No. Excluding it from the primes is necessary to preserve the uniqueness of prime factorization.
Q. How do I quickly count the divisors of a number?
Factor it into primes, add 1 to each exponent, then multiply those results. 12 = 2²×3¹, so (2+1)(1+1) = 6 divisors.
Q. Why is 2 the only even prime?
Every even number larger than 2 is divisible by 2, so none of them can be prime.