Divisibility Calculator
Check divisibility of a number by 2 through 12 and find all factors. Enter values for instant results with step-by-step formulas.
Calculator
Adjust values & calculateDivisibility Rules (2-12)
Factor Pairs
Formula
A number n is divisible by d if dividing n by d leaves no remainder. The total number of factors of n = p1^a1 x p2^a2 x ... is (a1+1)(a2+1)... Factor pairs multiply together to give n, and factors up to sqrt(n) are sufficient to find all factors.
Last reviewed: December 2025
Worked Examples
Example 1: Divisibility Analysis of 360
Example 2: Factor Analysis of 84
Background & Theory
The Divisibility Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Divisibility Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Sources & References
Formula
n is divisible by d if n mod d = 0
A number n is divisible by d if dividing n by d leaves no remainder. The total number of factors of n = p1^a1 x p2^a2 x ... is (a1+1)(a2+1)... Factor pairs multiply together to give n, and factors up to sqrt(n) are sufficient to find all factors.
Worked Examples
Example 1: Divisibility Analysis of 360
Problem: Check divisibility of 360 by 2 through 12 and find all factors.
Solution: 360 / 2 = 180 (divisible, last digit 0 is even)\n360 / 3 = 120 (divisible, digit sum 3+6+0=9 is divisible by 3)\n360 / 4 = 90 (divisible, last two digits 60 divisible by 4)\n360 / 5 = 72 (divisible, last digit is 0)\n360 / 6 = 60 (divisible, divisible by both 2 and 3)\n360 / 7 = 51.43 (NOT divisible)\n360 / 8 = 45 (divisible)\n360 / 9 = 40 (divisible, digit sum 9 divisible by 9)\n360 / 10 = 36 (divisible)\n360 / 12 = 30 (divisible)\nPrime factorization: 2^3 x 3^2 x 5
Result: 360 has 24 factors | Divisible by 2,3,4,5,6,8,9,10,12 | Not by 7,11
Example 2: Factor Analysis of 84
Problem: Find all factors, factor pairs, and prime factorization of 84.
Solution: Check 1 to sqrt(84) = 9.17:\n84/1=84, 84/2=42, 84/3=28, 84/4=21, 84/6=14, 84/7=12\nFactors: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84\nFactor pairs: (1,84), (2,42), (3,28), (4,21), (6,14), (7,12)\nPrime factorization: 2^2 x 3 x 7\nNumber of factors: (2+1)(1+1)(1+1) = 12
Result: 84 has 12 factors | 84 = 2^2 x 3 x 7 | Divisible by 2,3,4,6,7,12
Frequently Asked Questions
What are divisibility rules and why are they useful?
Divisibility rules are shortcut tests that allow you to determine whether a number is divisible by another number without performing full division. These rules exploit patterns in the decimal number system to make quick mental calculations possible. For example, a number is divisible by 2 if its last digit is even, and divisible by 3 if the sum of its digits is divisible by 3. These rules are essential for simplifying fractions, factoring numbers, finding GCF and LCM, and checking arithmetic. Before calculators were widely available, these rules were primary tools for number manipulation. They remain valuable today for mental math, number theory education, and building intuition about numerical patterns.
What is the divisibility rule for 7 and why is it more complex?
The divisibility rule for 7 involves taking the last digit, doubling it, and subtracting from the remaining digits. If the result is divisible by 7, so is the original number. For example, for 371: double the last digit (1 x 2 = 2), subtract from the remaining number (37 - 2 = 35), and 35 is divisible by 7, so 371 is divisible by 7. This rule is more complex than others because 7 does not have a simple relationship with powers of 10. The rule works because 10 is congruent to 3 modulo 7, and 21 (3 times 7) provides the doubling relationship. An alternative rule is to multiply the last digit by 5 and add to the remaining digits, which some find more intuitive.
What is a perfect number and how does it relate to divisibility?
A perfect number is a positive integer that equals the sum of its proper divisors (all divisors except itself). The first perfect number is 6, whose proper divisors are 1, 2, and 3, which sum to 6. The next perfect numbers are 28, 496, and 8128. Euclid proved that if 2^p - 1 is prime (a Mersenne prime), then 2^(p-1) times (2^p - 1) is a perfect number. Euler later proved that every even perfect number has this form. Whether odd perfect numbers exist is one of the oldest unsolved problems in mathematics. Numbers whose proper divisors sum to less than the number are called deficient, while those that sum to more are called abundant. For example, 12 is abundant because 1+2+3+4+6 = 16 > 12.
How does the divisibility rule for 11 work?
The divisibility rule for 11 uses the alternating sum of digits: subtract and add digits alternately from right to left (or left to right), and if the result is divisible by 11 (including zero), the original number is divisible by 11. For example, for 918082: 9 - 1 + 8 - 0 + 8 - 2 = 22, which is divisible by 11, so 918082 is divisible by 11. This rule works because powers of 10 alternate between leaving remainders of 1 and 10 when divided by 11, and 10 is congruent to -1 modulo 11. This creates the alternating pattern. The rule is particularly elegant and can be applied quickly for any number of digits, making it one of the more practical divisibility rules for mental arithmetic.
How does divisibility relate to modular arithmetic?
Divisibility and modular arithmetic are deeply connected. Saying that a divides b is equivalent to saying b is congruent to 0 modulo a, or b mod a = 0. Divisibility rules are essentially techniques for computing remainders in the decimal system without performing full division. The digit sum rule for 3 works because 10 is congruent to 1 mod 3, so each digit contributes its face value to the remainder. The alternating sum rule for 11 works because 10 is congruent to -1 mod 11. Understanding these connections provides deeper insight into why divisibility rules work and allows you to derive rules for any divisor. Modular arithmetic generalizes divisibility to a complete arithmetic system used in cryptography, computer science, and abstract algebra.
What practical applications use divisibility in everyday life?
Divisibility has numerous everyday applications that people often use without realizing it. When splitting a restaurant bill evenly among friends, you are checking divisibility. When determining how many rows of tiles fit a bathroom floor without cutting, you need the dimensions to be divisible by the tile size. Packaging products into boxes of equal size requires divisibility. In cooking, doubling or halving recipes involves divisibility of ingredient quantities. In music, time signatures and beat subdivisions rely on divisibility for rhythm patterns. Calendar calculations use divisibility rules for leap years (divisible by 4 but not 100, unless also divisible by 400). Even barcode check digits use modular arithmetic and divisibility to detect scanning errors.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy