Number Theory
Prime Number Checker
Check whether a number is prime instantly (up to 24 digits), see its smallest factor and the nearest primes.
A prime has exactly two divisors: 1 and i…
Also called is it prime · prime checker
Math · 10 calculators
Number Theory Calculators
Primes, factorisation, GCD and LCM, divisors, Fibonacci numbers, perfect numbers and modular arithmetic.
Number Theory
Prime Factorization Calculator
Break any whole number (up to 18 digits) into its prime factors, shown in exponent form with a factor tree and divisor count.
n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ
Also called prime factors · factor tree
Number Theory
GCD / HCF Calculator
Find the greatest common divisor (GCD), also called the highest common factor (HCF) or GCF, with Euclidean algorithm steps.
Euclid: GCD(a, b) = GCD(b, a mod b), GCD(…
Also called hcf calculator · greatest common divisor
Number Theory
LCM Calculator
Find the least common multiple (LCM) of two or more whole numbers, with prime factorisations and steps.
LCM(a, b) = |a × b| ÷ GCD(a, b)
Also called least common multiple · lowest common multiple
Number Theory
Factors Calculator
List every factor (divisor) of a number, factor pairs, the count and sum of divisors, and whether the number is prime.
d is a factor of n if n ÷ d leaves no rem…
Also called divisor calculator · factors of a number
Number Theory
Fibonacci Calculator
Find the nth Fibonacci number exactly (up to n = 5,000) and list the sequence, with the golden-ratio approximation.
F(0) = 0, F(1) = 1, F(n) = F(n−1) + F(n−2)
Also called fibonacci sequence · nth fibonacci number
Number Theory
Perfect Number Checker
Check if a number is perfect, abundant or deficient by comparing it with the sum of its proper divisors.
Perfect: σ(n) − n = n
Also called is perfect number · abundant number
Number Theory
Even or Odd Checker
Instantly check whether any integer — even a very long one — is even or odd, with the reason.
Even if n mod 2 = 0, otherwise odd
Also called is it even or odd · odd or even
Number Theory
Modular Arithmetic Calculator
Modular exponentiation (aᵇ mod m), modular inverse and a mod m for very large integers using BigInt precision.
aᵇ mod m via square-and-multiply
Also called modular exponentiation · modular inverse calculator
Number Theory
Remainder Calculator
Divide two numbers and get the quotient and remainder (long division result), e.g. 17 ÷ 5 = 3 R 2.
Dividend = Divisor × Quotient + Remainder
Also called division with remainder · long division calculator
Key number theory formulas
| Prime Number Checker | A prime has exactly two divisors: 1 and itself |
|---|---|
| Prime Factorization | n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ |
| GCD / HCF | Euclid: GCD(a, b) = GCD(b, a mod b), GCD(a, 0) = a |
| LCM | LCM(a, b) = |a × b| ÷ GCD(a, b) |
| Factors | d is a factor of n if n ÷ d leaves no remainder |
| Fibonacci | F(0) = 0, F(1) = 1, F(n) = F(n−1) + F(n−2) |
| Perfect Number Checker | Perfect: σ(n) − n = n |
| Even or Odd Checker | Even if n mod 2 = 0, otherwise odd |
| Modular Arithmetic | aᵇ mod m via square-and-multiply · a⁻¹: a·x ≡ 1 (mod m) |
| Remainder | Dividend = Divisor × Quotient + Remainder |
Number Theory questions
Is 1 a prime number?
No. A prime must have exactly two distinct divisors; 1 has only one.
How do you find prime factors?
Divide by the smallest prime (2) as many times as possible, then 3, 5, 7 and so on until the quotient is 1. 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
Are GCD, HCF and GCF the same thing?
Yes. Greatest common divisor, highest common factor and greatest common factor are three names for the largest whole number that divides every number in the set.
What is the LCM used for?
The LCM is the smallest number that all given numbers divide into. It is used to find common denominators when adding fractions and to solve scheduling problems (when will two cycles line up again).
How is Fibonacci related to the golden ratio?
The ratio of consecutive Fibonacci numbers F(n+1)/F(n) approaches the golden ratio φ ≈ 1.618 as n grows.