Direct answer: To find the factors of a number, divide it by every whole number starting from 1 and keep only the divisions that come out even, with no remainder. Each exact divisor is a factor, and so is its partner quotient. You only need to test divisors up to the number’s square root — for 36, that means testing 1 through 6.
If you have ever stared at a homework question like “list all the factors of 36” and wondered where to start, you are not alone. Finding factors is one of the first skills students meet in math, and it shows up again everywhere: simplifying fractions, finding the GCF, solving prime factorization problems, and even in algebra later on. The good news is that the method is simple, and with one clever shortcut you can find every factor of a number in under a minute.
In this guide, you will learn the exact step-by-step method, the square-root shortcut that cuts your work in half, and two fully worked examples. And if you want to check your answers, here is how to find factors of a number instantly with Calcrio’s free tool — it finds the factors of a number, factor pairs, prime factorization, and the GCF and LCM.
What Is a Factor of a Number?
A factor is a whole number that divides another number exactly, leaving no remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 with nothing left over. But 5 is not a factor of 12, because 12 ÷ 5 = 2 remainder 2.
Here are three facts worth remembering:
- 1 is a factor of every number. Every whole number divides by 1.
- Every number is a factor of itself. 36 ÷ 36 = 1, so 36 is a factor of 36.
- Factors always come in pairs. If 4 is a factor of 36, its partner is 9, because 4 × 9 = 36. That pairing is the key to the whole method.
One common confusion: factors are not the same as multiples. The factors of 12 are 1, 2, 3, 4, 6, and 12 — a short, fixed list. The multiples of 12 go on forever: 12, 24, 36, 48, 60, and so on. Factors divide into a number; multiples are what you get when you multiply a number up.
How to Find Factors of a Number: The Step-by-Step Method
Follow these five steps for any whole number:
- Start with 1. Divide the number by 1. The result is the number itself, so 1 and the number are your first factor pair.
- Try 2, then 3, then 4, and so on. Divide the number by each whole number in turn. If the division is exact — no remainder — you have found two more factors: the divisor and the quotient.
- Skip the divisions with remainders. If a division leaves a remainder, that number is not a factor. Cross it out and move on.
- Stop at the square root. Once your test divisor passes the square root, you will only repeat pairs you already found.
- List the factors in order, smallest to largest. Collect every divisor and every quotient you found, sort them, and you have the complete list.
That is the whole method. Now let us see it in action with the number students are asked about most.
Worked Example: What Are the Factors of 36?
The number 36 is a perfect square: √36 = 6. That means we only need to test the divisors 1 through 6 — six quick divisions instead of thirty-six.
| Test divisor | Division | Exact? | Factor pair found |
|---|---|---|---|
| 1 | 36 ÷ 1 = 36 | Yes | (1, 36) |
| 2 | 36 ÷ 2 = 18 | Yes | (2, 18) |
| 3 | 36 ÷ 3 = 12 | Yes | (3, 12) |
| 4 | 36 ÷ 4 = 9 | Yes | (4, 9) |
| 5 | 36 ÷ 5 = 7.2 | No | — |
| 6 | 36 ÷ 6 = 6 | Yes | (6, 6) |
At 6, the pair meets itself: 6 × 6 = 36. That is your signal to stop — you have found them all.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36 — nine factors in total.
Notice that 5 failed. That is normal and expected: not every number up to 36 is a factor. The complete list has just 9 entries, and the square-root shortcut found them with only 6 divisions. You can double-check the answer with Calcrio’s free tool, which lists all factors of a number plus its factor pairs in one step.
The Square-Root Shortcut: Why You Only Test Up to √n
Here is the idea that makes the method fast. Factors come in pairs: in (2, 18), 2 is small and 18 is large; in (4, 9), 4 and 9 are closer. As you test larger divisors, the partners get smaller — until they meet in the middle at the square root.
For 36, the square root is 6, and 6 × 6 = 36. Once you reach 6, every divisor you test next would only repeat a pair you already found: testing 9 would give you back (9, 4), testing 12 would give (12, 3), and so on. So you stop.
The rule works for any number:
- For 24: √24 is about 4.9, so test 1 through 5 — the first whole number above the square root. That is five divisions, and the pairs you find already give you the large factors.
- For 100: √100 = 10, so test 1 through 10 — ten divisions instead of one hundred.
- For 72: √72 is about 8.5, so test 1 through 8.
This is the shortcut most ranking guides bury in a footnote. Use it first, and finding factors becomes a two-minute job.
Worked Example: What Are the Factors of 24?
Let us run the full method on 24, step by step.
| Test divisor | Division | Exact? | Factor pair found |
|---|---|---|---|
| 1 | 24 ÷ 1 = 24 | Yes | (1, 24) |
| 2 | 24 ÷ 2 = 12 | Yes | (2, 12) |
| 3 | 24 ÷ 3 = 8 | Yes | (3, 8) |
| 4 | 24 ÷ 4 = 6 | Yes | (4, 6) |
| 5 | 24 ÷ 5 = 4.8 | No | — |
√24 ≈ 4.9, so after testing 5 we are past the square root and done.
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 — eight factors in total.
Two numbers close in size can have very different factor counts: 36 = 2² × 3² gives (2 + 1) × (2 + 1) = 9 factors, while 24 = 2³ × 3 gives (3 + 1) × (1 + 1) = 8 factors. If that formula looks useful, our guide to prime factorization methods explains exactly where it comes from.
How Many Factors Does a Number Have?
You do not always need the full list — sometimes you just need the count. There is a shortcut for that too, and it comes straight from prime factorization:
- Break the number into prime factors with exponents. For 36: 36 = 2² × 3².
- Add 1 to each exponent: (2 + 1) and (2 + 1).
- Multiply the results: 3 × 3 = 9. So 36 has 9 factors.
Try it with a bigger number: 360 = 2³ × 3² × 5. Add 1 to each exponent and multiply: (3 + 1) × (2 + 1) × (1 + 1) = 4 × 3 × 2 = 24. So 360 has 24 factors — a fact you can verify in seconds with Calcrio’s free tool. This formula is a favorite of AI answer engines because it is short, numeric, and quotable — and it is the reason prime factorization matters.
3 Common Mistakes When Finding Factors
Mistake 1: Testing every number up to the target. Testing 1 through 36 to find the factors of 36 wastes time and invites errors. Stop at the square root.
Mistake 2: Listing only the small divisors. If you test 36 ÷ 2 = 18 and write down 2 but forget 18, your list is incomplete. Every exact division gives you two factors, not one.
Mistake 3: Confusing factors with multiples. The factors of 12 stop at 12. The multiples of 12 never stop. If your list keeps growing past the number itself, you are listing multiples.
Check Your Work with Calcrio’s Free Tool
The free tool at Calcrio takes any whole number and returns its complete factor list, its factor pairs, and its prime factorization. Enter two numbers and it also finds their GCF and LCM. It is free, with no sign-up — but learning the method above matters more, because the same skill powers fraction work, GCF and LCM problems, and algebra.
Once you can find factors by hand, the next steps in the cluster are natural: learn how to draw a factor tree to break numbers into primes, compare GCF vs LCM for two-number problems, and master factor pairs to see the rectangle picture behind every factor list.
Frequently Asked Questions
What is a factor of a number?
A factor is a whole number that divides another number exactly with no remainder. For example, 4 is a factor of 36 because 36 ÷ 4 = 9 exactly. Every number has 1 and itself as factors.
How do you find all the factors of a number?
Divide the number by 1, 2, 3, and so on up to its square root, keeping only the exact divisions. Each exact division gives you two factors: the divisor and the quotient. Then list them in order from smallest to largest.
What is the fastest way to find factors?
Use the square-root shortcut: test divisors only up to the square root of the number. For 36, that means testing just 1 through 6, since √36 = 6. The pairs you find along the way give you the rest automatically.
How many factors does 36 have?
36 has 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, and 36. You can confirm this with the counting formula: 36 = 2² × 3², so (2 + 1) × (2 + 1) = 9 factors.
What are the factors of 24?
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 — eight factors in total. They come in pairs: (1, 24), (2, 12), (3, 8), and (4, 6).
What is the difference between factors and multiples?
Factors divide into a number exactly, and the list is finite — the factors of 12 are 1, 2, 3, 4, 6, and 12. Multiples are what you get by multiplying a number upward, and they go on forever: 12, 24, 36, 48, 60, and so on.