🌱 Factors · multiples · primes
Primes Won’t Be Boxed
Most numbers grow into a neat rectangle. A stubborn few refuse, and those are the ones that matter.
Take twelve seeds. Lay them out in a tidy rectangle.
Three rows of four. Sorted.
Now try thirteen.
Shuffle them however you like: you get a leftover, or one long lonely line.
The puzzle
What’s a rectangle got to do with it?
Here’s the whole idea in one picture. Line seeds up into a full rectangle (no gaps, no leftovers) and the two sides are a pair of factors. 12 falls into a neat 3 × 4. But 13 won’t: whatever you try, you get a single miserable strip.
Your turn → Feed the machine any number up to 60. Tap the chips, or nudge it with −1 and +1, and watch it find every rectangle that number can make.
12
Some numbers look prime.
Odd. Awkward. In no times table you’ve ever chanted.
Most of the time, they’re lying.
The trap
Odd is not the same as prime
This is where nearly everybody gets caught. Pick one of these, then answer before the machine gets a go. No changing your mind after.
Is 51 prime?
Every rectangle has two sides.
So factors never arrive alone. They come holding hands.
Why pairs
Factors arrive holding hands
Every box the machine finds hands you two factors at once. 1×12, 2×6, 3×4… and then it mirrors. Pick a number and watch its factors pair off across the rainbow.
Take 12’s box, 3 × 4. Read it one way: 3 rows of 4. Now turn your head sideways: 4 rows of 3. Same dots, same box.
One rectangle, two ways of reading it, so every box you find hands you two factors in one go. That is all a factor pair is.
You never have to find a prime.
Sweep away everything that boxes. Whatever’s left standing is prime.
The sieve
Don’t hunt the primes. Sweep away everything else.
A Greek librarian called Eratosthenes worked this trick out over two thousand years ago, and nobody has beaten it since. Sweep away every number that boxes, and the primes are whatever’s left standing. Four sweeps clear the whole board to 60.
Why stop at 7? Because 8 × 8 is 64, already past 60. Any number up to 60 that boxes at all must have a side of 7 or less. And 1 sits the whole thing out: not prime, not composite, so the sieve doesn’t even glance at it.
An old friend
Sixty, the factor millionaire
Tap 60’s neighbours and count what each one owns.
Here’s the part that turns a trick into a law.
Every whole number past 1 is primes, multiplied together.
And only one way. 60 = 2 × 2 × 3 × 5. No other recipe will ever make 60.
Primes are the atoms of numbers. Everything else is built from them.
The whole number line is built out of things that won’t be boxed.
And that refusal isn’t only pretty. The codes guarding bank cards multiply two huge primes together (hundreds of digits each) because nobody can un-multiply the answer in a hurry. Primes keep secrets for a living.
Pass it on
Next time someone swears 51 is prime (or that 1 is), don’t argue. Draw them the box. Three rows of seventeen, done. If it boxes, it isn’t prime, and now you’re the one who can prove it.
Primes are the atoms; every other number is a molecule built from them. 60 = 2 × 2 × 3 × 5, forever.