Direct answer: Factor pairs are two whole numbers that multiply together to give a target number. For 36, they are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). To find them, test divisors from 1, write down each divisor with its partner quotient, and stop when the two numbers in a pair meet.
Factor pairs are the simplest idea in this whole topic — and the one that makes everything else click. Once you see that every factor has a partner, the factor list, the prime factorization, and even the GCF all fall into place. This guide shows you how to find factor pairs in three steps, the “stop when pairs meet” method, and how they lead straight to common factors and the GCF.
You can check any answer with Calcrio’s free tool, which lists the factor pairs of any number instantly, along with its full factor list, prime factorization, and the GCF and LCM of two numbers.
What Are Factor Pairs?
A factor pair is a pair of whole numbers that multiply together to make a given number. Every factor belongs to exactly one pair, and the two numbers in the pair are called partners.
For 12, the factor pairs are:
- 1 × 12 = 12 → (1, 12)
- 2 × 6 = 12 → (2, 6)
- 3 × 4 = 12 → (3, 4)
That is all of them — three pairs, six factors. The pairs are written with the smaller number first, and (2, 6) and (6, 2) are the same pair, not two different ones. Never list a pair twice in the other order.
For 36, the factor pairs are:
- 1 × 36 = 36 → (1, 36)
- 2 × 18 = 36 → (2, 18)
- 3 × 12 = 36 → (3, 12)
- 4 × 9 = 36 → (4, 9)
- 6 × 6 = 36 → (6, 6)
Five pairs, nine factors. The pair (6, 6) is special: both numbers are the same, which happens only when the number is a perfect square. That pair is your signal to stop — more on that below.
How to Find Factor Pairs in 3 Steps
The method is short enough to memorize:
- Start at 1 and test each number in order. Divide the target number by 1, then 2, then 3, and so on. If the division is exact, you have found a pair: the divisor and the quotient.
- Write each pair down once. Record (divisor, quotient) with the smaller number first. Skip any divisor that leaves a remainder — it is not a factor, so it has no pair.
- Stop when the pairs meet. When the divisor and the quotient are the same number, or the divisor passes the square root, you are done — every pair after that would just repeat one you already found.
Worked Example: Factor Pairs of 36
Run the method on 36, the number students are asked about most:
| Test divisor | Division | Pair found |
|---|---|---|
| 1 | 36 ÷ 1 = 36 | (1, 36) |
| 2 | 36 ÷ 2 = 18 | (2, 18) |
| 3 | 36 ÷ 3 = 12 | (3, 12) |
| 4 | 36 ÷ 4 = 9 | (4, 9) |
| 5 | 36 ÷ 5 = 7.2 | Not a factor — skip |
| 6 | 36 ÷ 6 = 6 | (6, 6) — stop here |
The factor pairs of 36 are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6).
At 6, the pair meets itself: 6 × 6 = 36. That is the square root of 36, and it is the natural stopping point. Testing 9 next would give you (9, 4) — the pair (4, 9) backwards. Nothing new lives past the square root.
Worked Example: Factor Pairs of 12
12 is smaller and faster:
| Test divisor | Division | Pair found |
|---|---|---|
| 1 | 12 ÷ 1 = 12 | (1, 12) |
| 2 | 12 ÷ 2 = 6 | (2, 6) |
| 3 | 12 ÷ 3 = 4 | (3, 4) |
| 4 | 12 ÷ 4 = 3 | Already found as (3, 4) — stop |
The factor pairs of 12 are (1, 12), (2, 6), and (3, 4).
Here the pairs meet without a perfect square: 12 is not a square number, so no pair has two equal numbers. Instead, you stop when the divisor (4) has already appeared as the second number of a pair you found (3, 4). The rule is the same in both cases: stop when the pairs meet.
The Rectangle Picture: Why Factor Pairs Matter
Factor pairs are not just a list — they are a picture. Every factor pair is the side lengths of a rectangle with the target number as its area.
For 12, the pairs give you three rectangles:
- 1 × 12: a long, thin strip, 1 unit by 12 units.
- 2 × 6: a rectangle 2 units by 6 units.
- 3 × 4: a rectangle 3 units by 4 units — the closest to a square.
For 36, the pairs give you five rectangles, ending in 6 × 6 — an actual square. A number has a square factor pair exactly when it is a perfect square.
This picture is why teachers draw arrays and rectangles when they teach factors: it turns an abstract list into something you can see, and it explains the “stop when pairs meet” rule — once the rectangle’s width passes its height, you are just drawing the same rectangles turned sideways.
It also explains a handy fact: the number of factor pairs is half the number of factors — except for perfect squares, where the (6, 6)-style pair counts once. 36 has 9 factors and 5 pairs; 12 has 6 factors and 3 pairs.
From Factor Pairs to Common Factors and the GCF
Factor pairs are the bridge to the next big skill: common factors and the GCF. Here is how the bridge works.
First, collect the factors from each number’s pairs:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
The factors that appear in both lists are the common factors: 1, 2, 3, and 6. The greatest of them is the GCF: 6.
The GCF of 12 and 18 is 6.
That is the whole connection: factor pairs give you the factor lists, the lists give you the common factors, and the biggest common factor is the GCF. The same route works for any pair of numbers: for 24 and 36, the common factors are 1, 2, 3, 4, 6, and 12 — so the GCF of 24 and 36 is 12. Our GCF vs LCM guide works through this in full, including the self-check formula GCF × LCM = product of the two numbers.
This is also why factor pairs are the first thing to learn before prime factorization: the pairs are the raw material. Our guide to how to find factors of a number covers the full method and the square-root shortcut, and the prime factorization guide shows how the pairs lead to prime factors like 36 = 2² × 3².
3 Common Mistakes with Factor Pairs
Mistake 1: Listing pairs twice. (2, 18) and (18, 2) are the same pair. List each pair once, smaller number first, and stop when the pairs meet — otherwise your list doubles for no reason.
Mistake 2: Missing the square pair. For 36, the pair (6, 6) is easy to forget because both numbers are the same. But 6 × 6 = 36, and without it your factor list has only 8 entries instead of 9. Perfect squares always have a pair where both numbers are equal.
Mistake 3: Confusing factor pairs with multiples. The factor pairs of 12 are (1, 12), (2, 6), and (3, 4) — three pairs, and the list ends. If your pairs keep going past 12, you are listing multiples, not factors. A factor pair always multiplies to exactly the target number.
What About Negative Factor Pairs?
Can factor pairs be negative? Yes — a negative times a negative is a positive, so every positive factor pair has a negative twin: for 12, (−1, −12), (−2, −6), and (−3, −4) all multiply to 12.
For school math up to Grade 8, you will almost always work with positive factor pairs only. But if a teacher asks for “all factor pairs,” the negative ones exist — and knowing that is the difference between a complete answer and a half answer.
Check Your Answer with Calcrio’s Free Tool
The free tool at Calcrio lists the factor pairs of any number, along with its complete list of factors of a number, its prime factorization, and the GCF and LCM of two numbers. It is free and instant. But the method above is what matters for the test: test divisors from 1, write each exact pair once, and stop when the pairs meet.
Once factor pairs feel easy, the next step is factor trees — the visual way to turn a number into its prime factors, branch by branch.
Frequently Asked Questions
What are factor pairs?
Factor pairs are two whole numbers that multiply together to give a target number. For 36, the factor pairs are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). Each pair multiplies to exactly 36.
How do you find factor pairs of a number?
Test divisors starting from 1 and write down each exact division as a pair: the divisor and its quotient. For 36, testing 1 through 6 gives the pairs (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). Stop when the two numbers in a pair meet.
What are the factor pairs of 12?
The factor pairs of 12 are (1, 12), (2, 6), and (3, 4). Check: 1 × 12 = 12, 2 × 6 = 12, and 3 × 4 = 12. There are three pairs and six factors in total.
What are the factor pairs of 36?
The factor pairs of 36 are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). There are five pairs and nine factors. The pair (6, 6) appears because 36 is a perfect square: 6 × 6 = 36.
How do factor pairs help find the GCF?
Factor pairs give you the factor list of each number. The factors both numbers share are the common factors, and the greatest of them is the GCF. For 12 and 18, the common factors are 1, 2, 3, and 6 — so the GCF is 6.
What is a common factor?
A common factor is a number that divides two or more numbers evenly. For 12 and 18, the common factors are 1, 2, 3, and 6. The greatest common factor — the GCF — is 6.