Factor Calculator - All Factors and Factor Pairs

A factor (also called a positive divisor) of a whole number n is a positive integer that divides n with no remainder. The factors of 36 are every integer d such that 36 ÷ d is also an integer: 1, 2, 3, 4, 6, 9, 12, 18, and 36. A factor pair is two factors that multiply back to n, written here with the smaller factor first: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6).

This free online factor calculator is an all-factors tool. It does not stop at the prime factorization 2² × 3². Those two views answer different questions. “What tiles a 36-square rectangle?” wants the full divisor list. “What is 36 as a product of primes?” wants the prime factorization calculator. If you already have several integers and want the divisors they share, use the greatest common factor calculator or the least common multiple calculator.

The search is capped at n = 10¹². The algorithm walks i from 1 through √n and records both i and n/i whenever i divides n. For n ≤ 10⁹ that loop is short; for n up to 10¹² it is about a million trial divisions, which a browser can finish without a worker thread. n = 1 is allowed and returns the single factor [1]. Zero, negatives, and non-integers are rejected because the school definition of “list the factors of n” assumes a positive integer.

Primality is read off the factor count: exactly two distinct positive factors means prime. Nine factors means 36 is composite. One factor means n = 1, which is neither.

Using this divisor lister is straightforward:

1. Type a positive integer n between 1 and 1,000,000,000,000. Commas are ignored. The default is 36.
2. Read the sorted factor chips and the pair table (a, n/a) with a ≤ n/a. A square number has a middle pair where a = n/a.
3. Check the count and the prime / composite label. Copy the full text summary, or clear the field.

Note: the last n is stored in localStorage for up to 30 days on this device only. Nothing is uploaded.

Walking all the way to n is wasted work. Every factor larger than √n is the complement of a factor already found. For 36, √36 = 6, so the loop is 1 through 6:

• 1 divides 36 → pair (1, 36)
• 2 divides 36 → pair (2, 18)
• 3 divides 36 → pair (3, 12)
• 4 divides 36 → pair (4, 9)
• 5 does not divide 36
• 6 divides 36 → pair (6, 6), the square-root pair

Sort the unique values and you have 1, 2, 3, 4, 6, 9, 12, 18, 36. The same skeleton works for a prime such as 37: the only hits are 1 (with complement 37) and nothing else through floor(√37) = 6, so the factor list is [1, 37] and the number is labeled prime.

Highly composite n near 10¹² still have only a modest number of divisors - hundreds, not millions. The expensive part is the trial loop, not printing the list. That is why this page lists every divisor up to the cap instead of skipping the listing past 10⁹.

Pairs make the multiplication table visible. Teachers often ask students to “list factor pairs of 36” before they ask for a prime-power product. A perfect square is exactly the case where one pair is a repeated factor: (6, 6) for 36, (1, 1) for 1, (12, 12) for 144. That repeated pair is why a square has an odd number of distinct factors: nine for 36, three for 9, fifteen for 144.

A non-square always has an even factor count because every divisor has a different complement. 10 has four factors; 12 has six; 7 has two. You can therefore test “is n a perfect square?” by checking whether the factor count is odd, which this page makes obvious without a separate square-root widget.

The factor-count definition is the one this calculator uses because it does not require naming primes in advance:

• Prime: exactly two distinct positive factors (1 and itself).
• Composite: three or more distinct positive factors.
• Neither: n = 1, whose only factor is 1.

2 is the only even prime. 9 is composite even though it is odd. 1 is not prime: if it were, unique prime factorization would collapse, because you could multiply by 1 any number of times. This page will not call 1 prime, and it will not pretend 1 has a prime factorization other than the empty product.

A factor of n is at most n (for positive integers). A multiple of n is at least n and there are infinitely many: 36, 72, 108, … Factors of 36 live in a finite list; multiples of 36 do not. Mixing the words is the most common vocabulary error in this unit.

The greatest common factor of two numbers is the largest entry that appears in both factor lists. The least common multiple is the smallest positive integer that appears in both multiple lists. You do not need this page’s full list to compute either one - Euclid’s algorithm is faster - but seeing the lists for small n is how the definitions become believable. Jump to the GCF calculator or the LCM calculator when you have two or more integers.

Checking a factor list by hand is faster if you already know what “done” looks like. The following examples are the ones classrooms actually assign, plus one large-n check so the 10¹² cap is not abstract.

n = 1. The loop runs to √1 = 1 and records the pair (1, 1). The unique list is [1]. Count 1: neither prime nor composite. Every later definition that says “primes are the multiplicative atoms” depends on excluding 1.

n = 12. √12 ≈ 3.46, so test 1, 2, 3. Hits: (1, 12), (2, 6), (3, 4). Sorted factors: 1, 2, 3, 4, 6, 12. Six factors, even count, not a square. Composite. This is the rectangle list for a dozen eggs: 1×12, 2×6, 3×4.

n = 17. √17 ≈ 4.12. Tests 1–4: only 1 divides. Pair (1, 17). Two factors: prime. You never needed to “know the primes up to 17” first; the failed tests of 2, 3, and 4 were the proof.

n = 36 is the default: nine factors, five pairs including (6, 6). Odd count because it is 6². The prime factorization 2² × 3² is how you would generate that list without trial division: every factor is 2^a × 3^b with 0 ≤ a ≤ 2 and 0 ≤ b ≤ 2, which is 3 × 3 = 9 factors. Generating from primes is the inverse of this page’s job; this page recovers the list from n alone.

n = 100 = 10² = (2×5)². Factors: 1, 2, 4, 5, 10, 20, 25, 50, 100. Nine again (square). Pairs (1, 100), (2, 50), (4, 25), (5, 20), (10, 10). Percent problems that “break 100 into factors” are this list.

A larger check: n = 999,999,999,989 is past the cap and is rejected. Stay under 10¹². For n = 1,000,000,000,000 = 10¹² = (2×5)¹²the factor count is 13 × 13 = 169 (exponents 0 through 12 for 2 and for 5). The trial loop still only runs to 10⁶. If the count you get is not 169, a transcription error crept in. The divisor function τ(n) (tau) is exactly this page’s count; number-theory texts write τ(36) = 9, τ(p) = 2 for prime p, τ(p^k) = k+1.

A practical self-check for any n: the product of each pair must equal n, the list must be strictly increasing, 1 and n must appear, and if you claim a prime then nothing in 2 … floor(√n) may divide. If two consecutive even numbers both appear as factors of an odd n, you listed someone else’s homework.

Once you have every proper divisor (all factors except n itself), their sum classifies n. 6 = 1+2+3 is perfect; 12 has proper-divisor sum 1+2+3+4+6 = 16 > 12 and is abundant; 17 has proper-divisor sum 1 and is deficient, as all primes are. 36 has proper-divisor sum 1+2+3+4+6+9+12+18 = 55 > 36, so 36 is abundant. This calculator does not print the aliquot-sum label because the assignment is almost always “list the factors,” but the list you copied is enough to finish those extra questions without a second website.

Proper divisors also appear in Egyptian fraction writing and in leftover-sharing puzzles (“divide 36 cookies into equal groups with none left”). Those puzzles are factor-pair questions with extra words. Translate to “which pair (a, 36/a) matches the story,” then stop.

If n = p1^e1 × p2^e2 × …, every factor is obtained by choosing an exponent at most ei for each prime. The number of factors is (e1+1)(e2+1)…. For 36 = 2^2 × 3^2 that product is 3×3 = 9, matching the default list. For 12 = 2^2 × 3 the count is 3×2 = 6. For a prime p^k the count is k+1: 8 = 2^3 has factors 1, 2, 4, 8. This page does not ask you for the primes first. It recovers the list from n by trial division up to √n, which is the right direction when the question is “what are the factors?” rather than “factorize n.”

The two directions meet in homework that says “use prime factorization to find all factors of 72.” 72 = 2^3 × 3^2, so τ(72) = 4×3 = 12 factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Type 72 here to confirm. If a student lists eleven numbers, they dropped a pair - usually (8, 9) because 8 and 9 do not look related until you remember 8×9 = 72. The pair table on this page exists to catch that miss.

60 is another standard: 2^2 × 3 × 5, twelve factors (3×2×2). Pairs (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10). 60 is highly composite relative to its size, which is why it appears in clocks, degrees, and Babylonian leftovers. 7 is the opposite: two factors, one pair (1, 7), labeled prime. 49 = 7^2 has three factors 1, 7, 49 and the square pair (7, 7). Use those four numbers - 7, 49, 60, 72 - as a two-minute self-test after you read the default 36.

When n has three or more distinct primes the factor list gets long while √n stays moderate. 210 = 2×3×5×7 has sixteen factors; √210 is only about 14.5, so the loop is tiny. Do not confuse “many factors” with “slow”: speed follows √n, length follows the exponent product. That is why a trillion that is a high power of 10 still lists in a blink, and why this page can promise a full list up to 10^12 without a progress bar.