No rectangle but a line
Which Numbers Refuse to Split?
A prime number cannot be arranged into any rectangle except a single long row.
-
Step 1 of 5
Some counts can be arranged into rectangles.
Twelve counters make three rows of four, or two rows of six. Each arrangement is a way of splitting twelve into equal groups.
-
Step 2 of 5
Some counts stubbornly refuse.
Try the same with eleven and nothing works. Every attempt leaves a gap or a leftover, no matter which row length you choose.
-
Step 3 of 5
Those stubborn numbers are the primes.
A prime has no equal groupings apart from one long row, or a single column. It cannot be built from smaller whole numbers multiplied.
-
Step 4 of 5
Every other number is built from them.
Numbers that are not prime break down into primes, and always into the same ones. Twelve is two, two and three, however you start.
-
Step 5 of 5
They thin out, but never run out.
Primes get rarer as numbers grow, yet they never stop appearing. There is no largest one, a fact proved over two thousand years ago.
Euclid proved around 300 BC that primes never run out, and his argument is still the one taught today.
The short version
A prime number cannot be arranged into any rectangle but a single row, and every other number is built by multiplying primes together.
Try it yourself
Take 13 coins and try to lay them in equal rows. Only one row of 13 works, which is exactly what makes 13 prime.
Topics