Factor Pairs Calculator
A factor pair is two whole numbers that multiply to make a given number, like 3 times 4 for 12. Listing every factor pair is the clearest way to see all the ways a number can be split into equal groups, and it reveals all the divisors at once. This tool finds every factor pair of a positive whole number by testing divisors up to its square root, then reports the complete list of pairs, the number of pairs, and the total count of distinct factors. Perfect squares produce a repeated middle pair, giving an odd factor total.
How factor pairs are found
For each integer i from 1 to sqrt(N):
if N mod i = 0 then (i, N / i) is a factor pair
Factor count = number of distinct factors in all pairs
Here N is your number. Testing only up to the square root finds every pair exactly once, because the larger member of each pair is the quotient N divided by the smaller.
How factor pairs work
- A factor pair is two whole numbers whose product is the target number.
- Every factor of the number appears in exactly one pair.
- Testing divisors only up to the square root finds all pairs without duplication.
- A perfect square has a pair of equal factors, giving an odd total factor count.
- A prime number has exactly one factor pair: one times itself.
Factor pairs: frequently asked questions
What is a factor pair?
A factor pair of a whole number is a pair of whole numbers that multiply together to give that number. For 12 the factor pairs are 1 times 12, 2 times 6, and 3 times 4. Listing factor pairs is a tidy way to find all the divisors of a number and to understand how it can be split into equal groups.
How do you find all the factor pairs of a number?
Check each whole number from 1 up to the square root of the target. Whenever one divides the target exactly, it forms a pair with the quotient. For 36 you would test 1 through 6, finding 1 times 36, 2 times 18, 3 times 12, 4 times 9, and 6 times 6. Stopping at the square root avoids listing each pair twice.
Why does a perfect square have an odd number of factors?
A perfect square such as 36 has one factor pair where both numbers are the same, namely 6 times 6. That shared factor is counted once, so the total number of distinct factors is odd. Every other number has factors that come in distinct pairs, giving an even total.
Does this calculator handle prime numbers?
Yes. A prime number has exactly one factor pair: 1 times the number itself, because a prime has no divisors other than 1 and itself. The calculator will show that single pair and report a factor count of two, which is the defining property of a prime number.
What inputs are valid?
The calculator works on positive whole numbers (integers of one or more). It floors any decimal you enter and rejects zero and negatives, since factor pairs are defined for positive integers. To keep the page responsive, very large inputs are capped, and the cap is noted if you exceed it.
Official sources
- National Institute of Standards and Technology: Digital Library of Mathematical Functions: divisors and factorisation.
- National Aeronautics and Space Administration: Mathematics reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.