The divisors calculator above takes a whole number and returns every number that divides it exactly, the count of those divisors, the sum of all of them, the sum of the proper ones, and the classification that follows from comparing that sum to the number itself. It also gives the prime factorisation, because the factorisation is what all the other quantities are computed from once the numbers get large.
Arb Digital publishes free calculators for the arithmetic that is tedious rather than difficult. Listing divisors by hand is exactly that: entirely mechanical, and easy to leave one out — usually a divisor near the square root, or the number itself. Because a missed divisor changes the sum and can flip the classification, the tool builds the list by pairing, which makes omissions structurally impossible rather than merely unlikely.
What This Divisors Calculator Does
It trial-divides n by every integer up to the square root of n. Each divisor found below the root immediately supplies its partner above the root, so the list is complete by construction. The divisors are then sorted, counted and summed, and the number is classified by comparing the sum of its proper divisors against n.
Our prime factorization calculator is the adjacent tool and it does a narrower job: it breaks n into its prime factors and stops. That is one input to this page rather than a substitute for it, because the primes of 28 are 2, 2 and 7, while its divisors are 1, 2, 4, 7, 14 and 28. Our LCM and GCF calculator works with divisors shared between two numbers rather than the full divisor set of one, which is a different question again.
How to Use It
- Enter a positive whole number. Decimals and negatives are not meaningful here; divisibility is defined on integers.
- Choose a view. The pairs view is the clearest way to see the structure, because it shows each divisor beside the partner it multiplies with to give n.
- Read the count first. A number with few divisors relative to its size is close to prime; one with many is highly composite, and that distinction matters in most applications.
- Compare the two sums. Sigma of n includes n itself; s of n does not. The classification always uses the second one.
- Use the abundance figure. Zero is perfect, positive is abundant, negative is deficient, and the size of the number tells you how far from the boundary you are.
The Formulas and How They're Calculated
Both the count and the sum can be read straight off the prime factorisation, which is far faster than listing anything. If n = p₁a₁ × p₂a₂ × … then the number of divisors is d(n) = (a₁ + 1)(a₂ + 1)…, because a divisor is built by choosing any exponent from 0 up to aₕ for each prime independently. The sum of divisors is σ(n) = ∏ (pₕaₕ+1 − 1) ÷ (pₕ − 1), which is the product of a geometric series for each prime.
Take the default. 28 = 2² × 7, so d(28) = (2 + 1)(1 + 1) = 6, and the six divisors are 1, 2, 4, 7, 14 and 28. For the sum, the 2 part contributes 1 + 2 + 4 = 7 and the 7 part contributes 1 + 7 = 8, giving σ(28) = 7 × 8 = 56. Subtracting n leaves s(28) = 56 − 28 = 28, exactly equal to the number, so 28 is perfect and its abundance is zero. Wolfram MathWorld's entry on the divisor function gives both product formulas and the identities that follow from them.
That the two functions are multiplicative — that d(mn) = d(m)d(n) and σ(mn) = σ(m)σ(n) whenever m and n share no common factor — is the property doing the work. It is why the calculation splits cleanly across primes, and why a number's divisor structure is determined entirely by its exponents rather than by which primes happen to appear. The integer sequence of σ(n) values is catalogued as sequence A000203 in the OEIS.
Perfect, Abundant and Deficient Numbers
Compare s(n), the sum of the divisors below n, against n itself. If they are equal, n is perfect. If s(n) exceeds n, it is abundant. If s(n) falls short, it is deficient. The three categories cover every positive integer, and the classification is far from evenly split.
Deficient is the overwhelming default. Every prime is deficient in the most extreme way possible — its only proper divisor is 1, so s(p) = 1 regardless of how large p is. Every power of a prime is deficient too, and so is every divisor of a perfect number. Abundant numbers start at 12, whose proper divisors 1, 2, 3, 4 and 6 sum to 16, and they become common among numbers with many small prime factors. Multiples of an abundant number are themselves abundant, which is why abundance spreads once it appears.
Perfect numbers sit exactly on the boundary and are vanishingly rare: 6, 28, 496 and 8,128 are the only ones below ten thousand, and the next is 33,550,336. Euclid showed that 2p−1(2p − 1) is perfect whenever 2p − 1 is prime, and Euler proved that every even perfect number has that form, which ties them one-to-one to the Mersenne primes. Whether any odd perfect number exists is still unresolved after more than two thousand years. MathWorld's entry on perfect numbers sets out the Euclid-Euler result and the constraints established on any odd example.
Divisors Come in Pairs
Every divisor d of n has a partner n/d that is also a divisor, and the two multiply back to n. For 28 the pairs are 1 × 28, 2 × 14 and 4 × 7. This pairing is the reason you never have to test beyond the square root: one member of each pair is at or below √n, so finding all the small divisors hands you all the large ones for free.
It also explains a fact that surprises people the first time they meet it. Almost every number has an even count of divisors, because they pair off. The exceptions are the perfect squares, where one pair collapses — for 36, the pair 6 × 6 is a single divisor rather than two — so a number has an odd number of divisors if and only if it is a perfect square. That is the whole solution to the classic locker-door puzzle, and it also means the practical cost of factoring here is proportional to √n rather than n, which is why numbers into the billions still resolve instantly.
Why Divisor Counts Jump Around So Much
Divisor counts do not grow smoothly with size. The number 1,000 has 16 divisors while the much larger 1,009 has only 2, because it happens to be prime. What matters is not magnitude but the shape of the exponents in the factorisation.
Since d(n) is the product of one-more-than-each-exponent, several small primes with modest exponents beat one large prime every time. That is why 720 = 2⁴ × 3² × 5 has 5 × 3 × 2 = 30 divisors, more than any smaller number does. Such record-setters are called highly composite numbers, and the practical consequence is everywhere: 12 hours on a clock, 360 degrees in a circle, and 60 minutes in an hour were all chosen because those counts divide many ways without fractions. If you want to compare arrangements rather than factors, our permutation calculator counts orderings, and our factorial calculator handles the products that generate them.
Where Divisor Structure Actually Matters
Reducing a fraction to lowest terms is a divisor problem: you are looking for the largest number dividing both the numerator and the denominator. Laying tiles or splitting a grid without cutting anything requires a common divisor of both dimensions. Scheduling a task to run evenly across a cycle needs a divisor of the cycle length, which is why cron-style intervals of 5, 10, 15, 20 and 30 minutes work cleanly while 7 and 13 do not.
In computing, hash tables are conventionally sized at a prime precisely to minimise divisors, so that keys with a shared factor do not collide systematically. The same reasoning runs in the opposite direction for stripe sizes and buffer lengths, where a highly composite size divides evenly under many access patterns. And in modular arithmetic, whether an inverse exists at all depends on the two numbers sharing no divisor above 1 — the condition our modular exponentiation calculator relies on throughout.
The classification has a practical reading too. Because abundance is measured against the number itself rather than in absolute terms, it scales: 12 is abundant by 4, while 7,500 is abundant by 6,868. Comparing those raw figures says almost nothing, which is why number theorists usually work with the abundancy ratio σ(n)/n instead. A ratio of exactly 2 is perfect, above 2 is abundant, and below 2 is deficient — and for 28 that ratio is 56/28 = 2 exactly. Framed that way, the search for an odd perfect number becomes a search for an odd n whose divisor sum lands precisely on twice itself, which is a far more demanding condition than it first appears.
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See Our Web Design Work Talk To Our TeamCommon Mistakes to Avoid
- Leaving out 1 or n — both are divisors of every number, and dropping either changes σ(n) and can flip the classification.
- Confusing divisors with prime factors — 28 has two distinct prime factors but six divisors, and the two lists answer different questions.
- Using σ(n) to classify — the comparison uses the sum of the proper divisors, which excludes n itself. Using the full sum makes every number look abundant.
- Stopping the search too early — trial division must run to the square root inclusive, and stopping just below it misses the middle divisor of a perfect square.
- Assuming bigger numbers have more divisors — a large prime has exactly two, while a much smaller highly composite number can have dozens.
Related Free Tools From Arb Digital
Break a number into primes with the prime factorization calculator, find shared factors and multiples with the LCM and GCF calculator, compute large products with the factorial calculator, count arrangements with the permutation calculator, or work with powers in modular arithmetic using the modular exponentiation calculator. The full free online tools hub lists every mathematics tool we publish.
Frequently Asked Questions
A divisor is any whole number that divides it exactly, leaving no remainder. Every positive integer has at least two, namely 1 and itself, and a prime has exactly those two.
Add one to each exponent in its prime factorisation and multiply the results. Since 28 is 2 squared times 7, it has three times two, which is six divisors.
Its proper divisors, meaning all the divisors below the number itself, add up to exactly the number. The smallest examples are 6, 28, 496 and 8128.
Abundant means the proper divisors sum to more than the number, as with 12, whose divisors 1, 2, 3, 4 and 6 total 16. Deficient means they sum to less, which is true of every prime and every prime power.
Divisors normally pair up, one below the square root and one above. Only in a perfect square does one pair collapse into a single value, so perfect squares are exactly the numbers with an odd divisor count.
In ordinary use yes, both mean a number that divides another exactly. Prime factors are the narrower set consisting only of the prime numbers in the factorisation.
Nobody knows. Every perfect number found so far is even and follows the Euclid-Euler form, and the question of whether an odd one exists is one of the oldest unsolved problems in mathematics.
