You have 36 tiles and want to lay them in a perfect rectangle, no cuts and no gaps. The possible layouts are 1 by 36, 2 by 18, 3 by 12, 4 by 9, and 6 by 6, and every one of those pairs comes from the factors of 36. A factor is simply a number that divides another number evenly, with nothing left over. This tool lists every factor of the number you enter, so questions like "what rectangles can I make" or "what divides this evenly" get answered in a click.
What counts as a factor
A factor of n is any whole number a such that some whole number b makes the multiplication work out exactly. In other words, a x b = n with no remainder and no fractions.
Formula: a is a factor of n when n = a x b with no remainder, where a and b are whole numbers
Take 36: 36 = 6 x 6, so 6 is a factor. Also 36 = 4 x 9, so both 4 and 9 are factors. The tool finds every such pair by testing each candidate up to the square root of n, which is enough, because any factor larger than the square root pairs with one smaller than it.
How to use it
- Type the whole number you want to break down.
- Submit it.
- Read the full list of factors, usually shown smallest to largest, often with their pairings.
Once you have the factors, the combination calculator can help with grouping problems that build on divisibility, and the factorial calculator sits nearby for the next step up in number theory.
Worked example
Find all the factors of 36. Start from 1 and test: 36 / 1 = 36, so 1 and 36 are a pair. 36 / 2 = 18, so 2 and 18 join the list. 36 / 3 = 12, adding 3 and 12. 36 / 4 = 9, adding 4 and 9. 36 / 5 does not divide evenly, so skip it. 36 / 6 = 6, adding 6 (it pairs with itself). Past the square root of 36, which is 6, every pair would just repeat what you already found. The complete set is 1, 2, 3, 4, 6, 9, 12, 18, 36. Count them: nine factors, and every tile layout from the opening example corresponds to one of those pairs.
Practical uses
Beyond tile layouts, factors decide how items can be split into equal groups, which is the heart of scheduling, packaging, and resource planning. In math class they are the stepping stone to greatest common divisors, least common multiples, and fraction simplification. The permutation and combination tool also leans on divisibility thinking when counting arrangements, so factor fluency pays off there too.
Mistakes people make
- Forgetting 1 and the number itself. Every number has at least these two factors. Leaving them out is the most common slip.
- Mixing up factors and multiples. Factors divide into a number; multiples are made by multiplying it. The factors of 12 stop at 12, while its multiples keep going: 24, 36, 48.
- Listing only one from each pair. Finding that 4 divides 36 but forgetting 9 misses half the layouts.
What it does not do
This tool lists factors; it does not do prime factorization into a product of primes, though the factor list makes that easy to read off by hand. It is built for whole numbers, so decimal inputs need rounding to an integer first.