Learn by doing · Prime numbers
Every number is primes, multiplied.
Prime Factor Finder takes a whole number apart into its prime factors — by hand, your taps doing the work. Divide it down, split it into a tree, group the primes to make factors, lay it out in blocks, and use the same primes to reduce a fraction. Every picture on this page is a screenshot from the app itself — nothing redrawn.
The app, at a glance
Two modes, chosen with the tabs at the top. Factor a number is the main one: type any whole number from 2 to 1,000,000 and all four sections below work on that same number. Reduce a fraction is the second tab. The examples on this page use 60, 100, 72, 29, and the fraction 84/90.
One thing to know before you start: 1 has no prime factors. It isn’t prime, and it isn’t built from primes — that’s why numbers start at 2.
1. Divide it down to Prime factors
HelpDivide by a prime, then divide the answer, and keep going until nothing is left but 1. Tap a prime from the row of prime buttons — if it divides evenly, the step lands on the ladder. If it doesn’t, nothing breaks; the app tells you there’s a remainder and you try another prime.
Two ways to write the answerPlain — 2 × 2 × 3 × 5 — and exponent form, 22 × 3 × 5. The app then multiplies the factors back as a check that they equal your number.
Bigger primes, and second thoughtsWhen what’s left is a prime bigger than 97, a “Try a bigger prime” box lets you divide by it — or finish with it as the last factor. Undo and Start over are always there.
2. Build a prime factor tree
HelpThe same factoring, drawn as a tree. Tap any number that isn’t prime and split it into two factors — a line connects it to the two numbers it splits into, like 60 down to 2 and 30. Keep splitting until every branch ends in a prime. Those ending numbers, in the green circles, are the prime factors.
Your splits, your treeAny split works — 60 as 2 × 30, 6 × 10, or 5 × 12. Different trees, same primes at the ends. Try two different first splits and compare.
Primes don’t splitA circle that holds a prime just sits there — it can’t be tapped, because there’s nothing left to split. That’s the tree’s way of showing you’re done.
3. Group the primes, make factors
HelpPrime factors are only half the story — ordinary factors are hiding inside them. Multiply any bunch of the primes together and the product is a factor of the whole number. Tap primes to build a group, make the group, and a box shows the result in large type, with the factor it pairs with.
Every factor, this wayThere’s no factor of a number that can’t be built by grouping its primes — grouping is how all of them are made. Leave one prime out and you’ve built the factor it pairs with.
The final boxWhen every prime is in a group, a gold box shows the whole breakdown and the app checks the group factors multiply back to your number.
4. Blocks — see the factors
HelpFor numbers up to 100, one more way to see factors: the number as blocks. Use the − and + buttons to change how many blocks go in each row. If every row comes out the same with nothing left over, that row size is a factor — leftovers show in red. The numbers across the top count the columns and the numbers down the left count the rows, so the size of the arrangement reads off at a glance.
Factor pairs you’ve foundEvery row size that comes out even adds its pair — 6 × 12 here — to the row of chips under the board. In a hurry? “Try every row size for me” tests them all at once.
Blocks has its own numberIt starts from your main number when that’s 100 or less, but you can type a different one just for the blocks — handy for hunting primes, next.
Discover primes: My prime collection
HelpBlocks also teach what prime means. If no row size fits — every arrangement leaves blocks over, except one row of all of them or all rows of one — the number can’t be split evenly at all. That’s a prime, and the app says so: congratulations, you found a prime number. Tap Add it to my prime collection and it joins your collection as a gold chip.
Proving itTry row sizes yourself — for most primes, two different sizes leaving leftovers is proof enough, and the app calls it. The smallest primes are quicker: 2 is called prime the moment its blocks show, because there’s no in-between size to test, and 3 is proven once its one in-between size — 2 — leaves a block over. “Try every row size for me” checks them all in one tap.
25 to find — then the board growsThere are 25 primes up to 100. The gold bar fills as you collect them, and at 25 of 25 the blocks board extends to numbers up to 200. The collection lasts while the page is open — reloading starts it over.
Did you know?
Multiply all 25 primes up to 100 together and you get 2,305,567,963,945,518,424,753,102,147,331,756,070 — a 37-digit number. Mathematicians call it the primorial of 100, and write it 100#.
It ends in just one zero. A trailing zero comes from a factor of 10, and 10 = 2 × 5 — the list of primes has exactly one 2 and one 5, so there’s only one 10 hiding in the multiplication.
Reduce a fraction
HelpThe second tab, and the payoff for all that factoring — now all on one line, with a heading over each of its three parts. Initial sits over the fraction you type. Prime Factors sits over the two rows of chips in the middle — the top number’s factors above the fraction bar, the bottom number’s below — both rows starting at the same left edge, with no labels to get in the way. Reduced sits over the reduced fraction at the end of the line. The line reads like an equation: your fraction = its prime factors = the reduced fraction. Tap a prime factor in the top row, then tap the same prime factor in the bottom row, and the pair cancels — both are struck through, and the reduced fraction at the end of the line changes right away.
The reduced fraction, liveThe reduced fraction sits at the end of the same line, in boxes like the ones you typed your fraction into, and it updates the moment each pair is removed. Cancel the 2s and 84/90 is already 42/45; cancel the 3s and it’s 14/15, in lowest terms. Cancelling is fair because it’s just dividing the top and the bottom by the same number, which never changes what the fraction is worth.
The greatest common factor, freeMultiply the cancelled factors together — 2 × 3 = 6 here — and you have the greatest common factor of the two numbers. Shortcuts if you want them: “Cancel all matching pairs” and “Put all the prime factors back”.
Pick a number and take it apart
Start with 60, the worked example on this page — or try a number that matters to you: a year, an age, the number of tiles in the box. Divide it, tree it, group it, block it.
Open Prime Factor Finder →Planned address: exponology.com/apps/factor-finder/app.html
Prime Factor Finder is the newest of the Exponology learning apps — the others work on American Mah Jongg hands and cryptogram puzzles. All of them share the same idea: you do the steps yourself, and the app keeps score.