Every factor of a number, listed in the pairs that produce them.
Every factor, in pairs
The pairing is the method.
A positive whole number. Commas are fine.
Factors of 36
9 factors
1, 2, 3, 4, 6, 9, 12, 18, 36
How many
9
(2 + 1) × (2 + 1) = 9
As primes
2² × 3²
the source of every factor above
Searched only to
6
past the square root, the pairs repeat
Proper factors add to
55
abundant
The pairs
Each pair of factors whose product is the number
1
×
36
2
×
18
3
×
12
4
×
9
6
×
6
the square root — counted once, not twice
Every factor below the square root has exactly one partner above it. That is why the search can stop at the root, and why the count is even unless the number is a perfect square.
Perfect, abundant or deficient
The factors below 36 add up to 55, which is more than 36. That makes it abundant; 12 is the smallest such number.
Only numbers up to 6 had to be tried, because factors come in pairs whose product is 36. Every factor above the square root is the partner of one below it, so searching past the root finds nothing new — it just meets the same pairs from the other end.
36 is a perfect square, so it has an ODD number of factors. Every other number has an even count, because the factors pair off — but 6 pairs with itself and is counted once.
The count from the exponents matches the length of the list — two routes, one number.
What this tool shows
Factors come in pairs whose product is the number. That pairing explains the count, explains why the search stops at the square root, and explains the one case where the count comes out odd.
Every factor, in ascending order
The pairs whose product is the number
The count derived from the prime exponents
Why perfect squares have an odd count
Perfect, abundant and deficient numbers
The sum of the divisors and Euler’s totient
Listed as pairs Count from the exponents, then verified Perfect, abundant or deficient The square-root bound explained
Complete lists; truncation is stated, never silent.
Updated 7 September 2026 · Works in any browser, no installation
The factors of a number are everything that divides it exactly, and they come in pairs whose product is the number. 36 pairs as 1×36, 2×18, 3×12, 4×9 and 6×6 — nine factors, because the last pair is 6 with itself and counts once. That is also why only numbers up to the square root have to be tested: everything above it is already someone’s partner.
At a glance
Formula shown
The number of divisors of n = p\u2081^e\u2081 \u00d7 \u2026 \u00d7 p\u2096^e\u2096 is (e\u2081 + 1) \u00d7 \u2026 \u00d7 (e\u2096 + 1). Divisors pair as d with n \u00f7 d, so only those up to \u221an need to be found.
Scenario support
Arranging a number of items into equal rows; finding common factors before reducing a fraction; checking whether a number of seats divides evenly into blocks.
Educational estimate
Planning support from the values you enter — not professional advice.
Factors come in pairs
If d divides n then so does n ÷ d, and the two go together. 1 pairs with the number itself, 2 with half of it, and so on inwards.
Listing factors as a flat row hides this completely. Written as pairs, two things become obvious at once: the count is even unless a pair happens to be a number with itself, and there is no point searching past the square root because everything beyond it has already appeared as a partner.
It is also the practical way to find them by hand. Go up from 1, and each hit gives you two. Stop when the two sides meet.
Counting them without listing them
45360 has exactly 100 factors, and you do not need to write any of them down to know that.
It is 2⁴ × 3⁴ × 5 × 7. Every factor is built by choosing how many 2s to include (five choices, 0 to 4), how many 3s (five), how many 5s (two) and how many 7s (two). 5 × 5 × 2 × 2 = 100.
The rule is: one more than each exponent, multiplied together. This page computes the count that way and separately counts the generated list, and the two are compared — if they ever disagreed the page would say so rather than showing one of them.
The odd one out
Every number has an even number of factors, except the perfect squares.
The reason follows straight from the pairing. Factors pair off, so they come two at a time — unless one pair is a number with itself, which happens exactly when the number is a square. 36 has 6 × 6, so 6 is counted once and the total is odd.
This is the whole content of the locker-doors puzzle, where a hundred lockers are toggled by a hundred students and the ones left open are the perfect squares. They are the lockers toggled an odd number of times, because only squares have an odd number of factors.
Perfect, abundant, deficient
Add up a number’s factors excluding itself. If the total is less than the number it is deficient, if more it is abundant, and if exactly equal it is perfect.
6 is perfect: 1 + 2 + 3 = 6. So is 28: 1 + 2 + 4 + 7 + 14 = 28. After that they get rare quickly — 496, then 8128, then 33550336. Only a few dozen are known at all.
Every even perfect number has the form 2^(p−1)(2^p − 1) with the second factor a Mersenne prime, which Euclid proved sufficient and Euler proved necessary. Whether any ODD perfect number exists is still open after more than two thousand years.
Most numbers are deficient. Every prime trivially is, since its only proper factor is 1.
Factor, divisor, multiple
Factor and divisor mean the same thing here: something that divides the number exactly. British schools tend to say factor, and number theory tends to say divisor, but nothing hangs on the difference.
Multiple is the other direction. 4 is a factor of 12; 12 is a multiple of 4. Getting these the wrong way round is the commonest confusion in the topic, and the test is simple: factors are never larger than the number, multiples are never smaller.
Prime factor is a factor that happens to be prime. 12 has six factors and only two prime ones, 2 and 3.
Where the list gets long
Highly composite numbers — those with more divisors than any smaller number — have a lot. 720 has 30 factors; 45360 has 100; 963761198400 has 6720.
They are why clocks and circles use 12, 60 and 360. Those numbers were chosen precisely because they divide many ways, which is a genuinely useful property for anything that has to be split into halves, thirds, quarters and fifths without fractions.
Where a list is too long to display, this page says so and shows the count anyway — the count comes from the exponents and does not need the list to exist.
Sources and methodology
Divisor counting and the classification into perfect, abundant and deficient go back to Nicomachus. These are the references.
Method. The number is factorised into primes by trial division to the square root, and the divisors are then generated from those exponents rather than by testing every candidate — which is why a hundred divisors appear instantly. The count is computed twice: once as the product of one-more-than-each-exponent, and once as the length of the generated list. The sum of the list is separately compared with sigma computed from the factorisation. That engine is verified on every change against 64 hand-written assertions, including that the exponent count, the list length and the summed list all agree for every number to three thousand, and that an odd count occurs exactly on the perfect squares. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Prime FactorizationBreak any number into primes with the division ladder shown, the number of trial divisions reported, and the argument for why the search can stop at the square root.
GCFThe greatest common factor of two to six numbers with all three routes shown — the shared primes to their lower powers, Euclid line by line, and the full factor lists when they are short enough to be honest.
Divisibility TestEvery divisibility rule from 2 to 16 applied to your number, each with its working, the reason it holds, and the true remainder beside it as a check.
Prime NumberWhether a number is prime, with a divisor named when it is not and the size of the search stated when it is. Deterministic, not probabilistic.
LCMThe least common multiple of two to six numbers, with the prime table that produces it, the multiples themselves, and how far counting would have had to go.
Simplify FractionsReduce a fraction to lowest terms with both routes to the divisor shown side by side — Euclid line by line and the shared primes — and a stated proof when nothing can be cancelled.
The primes underneath are on the Prime Factorization Calculator, which shows the ladder and the trial count. If you have two numbers and want what they share, the GCF Calculator does that directly rather than by intersecting two lists.
Educational use disclaimer
This is an educational tool. The list is complete and exact; where a number has more factors than can usefully be displayed the page says so rather than truncating quietly.
Published the factor page with the factors shown as PAIRS rather than a flat row, because the pairing is what explains both the count and the square-root bound.
A perfect square is the one case with an odd number of factors, and the page marks the pair that is a number with itself — the whole content of the locker-doors puzzle.
Each number is classified perfect, abundant or deficient from the sum of its proper factors, computed from the factorisation rather than by adding the list.
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