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.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Divisibility Calculator
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Formula: n is divisible by d if n mod d = 0
Worked example — 360 has 24 factors | Divisible by 2,3,4,5,6,8,9,10,12 | Not by 7,11
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) 360 / 3 = 120 (divisible, digit sum 3+6+0=9 is divisible by 3) 360 / 4 = 90 (divisible, last two digits 60 divisible by 4) 360 / 5 = 72 (divisible, last digit is 0) 360 / 6 = 60 (divisible, divisible by both 2 and 3) 360 / 7 = 51.43 (NOT divisible) 360 / 8 = 45 (divisible) 360 / 9 = 40 (divisible, digit sum 9 divisible by 9) 360 / 10 = 36 (divisible) 360 / 12 = 30 (divisible) Prime 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: 84/1=84, 84/2=42, 84/3=28, 84/4=21, 84/6=14, 84/7=12 Factors: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 Factor pairs: (1,84), (2,42), (3,28), (4,21), (6,14), (7,12) Prime factorization: 2^2 x 3 x 7 Number 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.
What are highly composite numbers and why are they special?
A highly composite number (HCN) is a positive integer that has more divisors than any smaller positive integer. The sequence begins 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, and continues. These numbers are special because they are optimally divisible, having the maximum number of factors for their size. This is why many measurement systems use them: there are 12 inches in a foot, 24 hours in a day, 60 minutes in an hour, and 360 degrees in a circle. Each of these values can be evenly divided by many numbers, making subdivision convenient. Highly composite numbers always have prime factorizations where the exponents are non-increasing, meaning they use small primes raised to higher powers before including larger primes.
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 for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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