Factor Pairs – Every Factor Has a Partner

When you divide a number by one of its factors, the answer is also a factor. These two factors that multiply together to give the original number are called a factor pair.

The idea of a factor pair is older than algebra itself. The Pythagoreans of ancient Greece (6th century BCE) thought about numbers geometrically, arranging pebbles into rectangular arrays — a number with several factor pairs could be laid out in several different rectangle shapes, while a number whose only pair is (1, itself) could ever only form a single line. That same picture is exactly why factor pairs matter today: any time you are laying out tiles, arranging seats into equal rows, or sizing a grid of pixels on a screen, you are really just choosing between the available factor pairs of the total.

What Is a Factor Pair?

A factor pair of N is two whole numbers (a, b) where a × b = N.

Finding All Factor Pairs

Factor pairs of 24

Factor a	Factor b	Check: a × b
1	24	24 ✓
2	12	24 ✓
3	8	24 ✓
4	6	24 ✓

√24 ≈ 4.9, so stop at 4. Factor pairs: (1,24), (2,12), (3,8), (4,6)

Factor pairs of 36

Factor a	Factor b	Check
1	36	✓
2	18	✓
3	12	✓
4	9	✓
6	6	✓ (perfect square pair)

Now build the pair table yourself for 48 — testing one divisor at a time, up to √48 ≈ 6.93, so only divisors 1 through 6 need checking.

Factor pair found

Not a factor (skipped)

Factor a

Factor b

√48 ≈ 6.93 → test divisors 1 to 6

Click Step 1 or Auto Play to begin!

Factor pairs of 48: (1,48), (2,24), (3,16), (4,12), (6,8)
Testing stopped at 6 because √48 ≈ 6.93 — any larger divisor would just repeat a pair already found.

Perfect Squares Have an Odd Factor

When a = b in a factor pair (like 6 × 6 = 36), the number is a perfect square. This gives it an odd number of factors because one factor (the square root) has no separate partner.

Number	Perfect Square?	Number of Factors
16	Yes (4×4)	5 (1, 2, 4, 8, 16)
12	No	6 (1, 2, 3, 4, 6, 12)
25	Yes (5×5)	3 (1, 5, 25)

Key Takeaways

A factor pair multiplies to give the original number.
Test divisors up to the square root; each gives a pair.
Perfect squares have one unpaired factor (the square root itself).
Factor pairs are especially useful when building factor trees.

Practice: Factor Pairs

Factor Pair Question