What are the factors of 36?
Use the Factor Calculator below — enter your number and click Calculate to list every factor.
What Is a Factor? Factors, Factor Pairs, Prime and Composite
A factor is a whole number that divides another number evenly, with nothing left over. If 12 ÷ 3 = 4 with no remainder, then both 3 and 4 are factors of 12. The complete list for 12 is 1, 2, 3, 4, 6 and 12 — six factors that can be paired up as factor pairs: 1 × 12, 2 × 6 and 3 × 4. Every pair multiplies back to the original number, which is why factors always come in pairs except for perfect squares, where the pair is the square root with itself (6 × 6 = 36).
Numbers fall into three groups by their factor count. A prime has exactly two factors, 1 and itself (7 has only 1 and 7). A composite number has more than two (12 has six). And 1 sits in its own category — neither prime nor composite, because it has just one factor. Understanding that split is the first step to prime factorization, the process of breaking any composite number down into primes that multiply back to it. Factoring a polynomial such as x² + 5x + 6 is a different job — that is what our trinomial factor calculator does.
How to Use This Factor Calculator (2026)
- Enter your number. Type any whole number from 1 up to 1 billion, or click a preset chip like 60 or 144.
- Click Calculate. You get the answer banner plus a full breakdown: every factor, every factor pair, the prime factorization in exponent form, and the count and sum of the factors.
- Read the steps. The working shows which divisors were tested up to the square root and how the perfect, abundant or deficient classification was decided — so you can repeat the method by hand on homework.
- Reset and try another. Reset clears the field for a fresh number; the results hide until you calculate again.
How to Find the Factors of a Number: 3 Methods
Method 1 — Division (the fastest by hand). Test whole numbers starting at 1 and stop as soon as your test number passes the square root of the target. Anything above the square root would only pair with something you have already found. For 60, √60 ≈ 7.75, so you test 1 through 7 only:
60 ÷ 2 = 30 → pair (2, 30) 60 ÷ 3 = 20 → pair (3, 20)
60 ÷ 4 = 15 → pair (4, 15) 60 ÷ 5 = 12 → pair (5, 12)
60 ÷ 6 = 10 → pair (6, 10) 60 ÷ 7 = 8 remainder 4 → stop
Method 2 — Factor pairs. Write the first and last numbers of a multiplication that lands on your target. Start with 1 × n, then 2 × n/2, and keep going until the two factors meet in the middle. This method makes patterns visible: pairs spread outward from the ends, which is why the factor list of 60 reads 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
Method 3 — Prime factorization. Divide the number by its smallest prime repeatedly until you reach 1, then collect the primes. For 60: 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1, so 60 = 2² × 3 × 5. Every factor of 60 can then be built by choosing how many 2s, 3s and 5s to include — the same trick the calculator uses to count factors without listing them. Break any number into primes with the full working in our prime factorization calculator.
Factors of Common Numbers — Reference Table
Quick lookup for the numbers that show up most in homework and tests. The factor count column comes from the divisor formula — raise each prime exponent by 1 and multiply — so for 120 = 2³ × 3 × 5 the count is (3+1)(1+1)(1+1) = 16.
| Number | All Factors | Count | Prime Factorization | Type |
|---|---|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 6 | 2² × 3 | Composite |
| 18 | 1, 2, 3, 6, 9, 18 | 6 | 2 × 3² | Composite |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 | 2³ × 3 | Composite |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 9 | 2² × 3² | Perfect square |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | 10 | 2&sup4; × 3 | Composite |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 12 | 2² × 3 × 5 | Composite |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 9 | 2² × 5² | Perfect square |
| 120 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 | 16 | 2³ × 3 × 5 | Composite |
| 97 | 1, 97 | 2 | 97 | Prime |
| 1 | 1 | 1 | — | Neither prime nor composite |
Why Factoring Matters: Real-World Uses
- Cryptography: RSA encryption depends on the difficulty of factoring a 2,048-bit number into two large primes — the same operation this calculator performs, scaled past any brute-force attempt.
- Sharing and splitting: the GCF answers “what is the largest equal group?” (24 apples and 36 oranges make piles of 12), and the LCM answers “when do two cycles line up?” (buses every 12 and 18 minutes both depart together at minute 36) — work both out in our GCF and LCM calculator.
- Cutting and tiling: carpenters and tilers use the LCM to find pattern repeats and minimise waste when two spacing intervals must realign.
- Growth and scaling: a growth factor of 1.05 per year is simply 5% a year — our percentage calculator converts between growth factors, decimals and percent.
- Fractions and ratios: every time a fraction is reduced to lowest terms, the numerator and denominator are being divided by their greatest common factor.
FAQ
A factor is a whole number that divides another number evenly, leaving no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12 because each of those divides 12 exactly. Every number has at least two factors, 1 and itself, and a number with exactly two factors is called a prime.
Test every whole number from 1 up to the square root of your number and pair each divisor with its match. For 60 the square root is about 7.7, so test 1 through 7: 60 ÷ 1 = 60, 60 ÷ 2 = 30, 60 ÷ 3 = 20, 60 ÷ 4 = 15, 60 ÷ 5 = 12 and 60 ÷ 6 = 10. That gives the pairs (1, 60), (2, 30), (3, 20), (4, 15), (5, 12) and (6, 10) for twelve factors in all. Stop as soon as the test number passes the square root — anything larger would only repeat a pair you already have.
Factors divide into a number evenly, while multiples are what you get by multiplying that number. The factors of 12 are 1, 2, 3, 4, 6 and 12, and the multiples of 12 are 12, 24, 36, 48 and so on. A positive factor of 12 is never larger than 12, and a multiple of 12 is never smaller than 12 — the two lists run in opposite directions from the same number.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36 — nine factors in all, paired up as 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. Its prime factorization is 2² × 3², and the divisor formula (2 + 1)(2 + 1) = 9 confirms the count. Because 36 = 6 × 6 it is a perfect square, so it has an odd number of factors.
The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100 — nine factors paired as 1 × 100, 2 × 50, 4 × 25, 5 × 20 and 10 × 10. Its prime factorization is 2² × 5², so (2 + 1)(2 + 1) = 9 factors in all. Like every perfect square, 100 has an odd factor count because the square root 10 pairs with itself.
A factor pair is two whole numbers that multiply to the same product — every factor of a number appears in exactly one pair. The factor pairs of 60 are (1, 60), (2, 30), (3, 20), (4, 15), (5, 12) and (6, 10), and the two numbers in each pair meet at or below the square root. Perfect squares are the exception: their middle pair is the square root twice, such as (6, 6) for 36.
Write the prime factorization n = p1a1 × p2a2 × …, then multiply (a1 + 1)(a2 + 1) and so on. Since 360 = 2³ × 3² × 5, it has (3 + 1)(2 + 1)(1 + 1) = 24 factors. A perfect square has an odd number of factors because one factor pair is the square root multiplied by itself: 36 has exactly 9 factors, while a prime has exactly 2.
Factoring underpins RSA encryption, which relies on how hard it is to factor a 2,048-bit number into two large primes. It also simplifies fractions with the greatest common factor, sets pattern repeats in tiling and cutting problems through the least common multiple, works out gear and pulley ratios, and scales recipes or material batches to the largest equal size. Any time something must divide evenly or realign on a cycle, factoring is the tool doing the work.
Sources: Common Core State Standards for Mathematics (4.OA.B.4, 6.NS.B.4), NGA Center and CCSSO; National Council of Teachers of Mathematics, Principles to Actions; D. M. Rosen, Elementary Number Theory; National Institute of Standards and Technology, RSA cryptosystem specifications. This tool is for educational purposes only.
Free Online Calculators
Factors at a Glance
| Number | # Factors | Prime Factorization |
|---|---|---|
| 24 | 8 | 2³ × 3 |
| 36 | 9 | 2² × 3² |
| 60 | 12 | 2² × 3 × 5 |
| 100 | 9 | 2² × 5² |
| 120 | 16 | 2³ × 3 × 5 |
| 97 | 2 | prime |
Quick Tips & Rules
- Only test divisors up to √n — everything above pairs with something below.
- GCF × LCM = a × b — one formula links both answers, worked through in the GCF and LCM calculator.
- Perfect squares have an odd number of factors.
- Reducing a fraction means dividing by the GCF — try the fraction calculator.
- Breaking a number into primes? Use the prime factorization calculator.
- Factoring x² + bx + c? The trinomial factor calculator does it step by step.