How to determine a number is a prime number or not

Is 91 prime? It looks stubborn. But the question is smaller than it looks. The numbers two to nine are all you need, and a handful of small divisions settles it.

Your body comes in pairs — two eyes, two hands. Yet the heart comes as one. Here's the whole road, in four steps. One: a prime has exactly two divisors.

Two: try dividing it by small numbers. Three, the shortcut — stop at the square root of n. Four: find a divisor and it's composite; find none, and it's prime. Follow those four steps and you have a recipe you can run on any number in seconds.

Let's watch multiplication build a number. Here's twelve, split every way it can be: one times twelve, two times six, three times four. Each row is a way to make twelve out of equal groups. Now take seven.

One times seven. That's the only row. Seven cannot be split into equal groups any other way. So here is the definition, a step at a time.

A prime is a whole number greater than one, whose only divisors are one and itself. Seven has exactly those two divisors, so seven is prime. Twelve has more, so twelve is composite. And one?

One has a single divisor, itself, so one is not prime. That means checking is a search: does any number from two up to n minus one divide n with nothing left over? If none does, n is prime. For 97, the honest search is long.

You'd try every number from 2 to 96 — that's the whole shaded stretch, and none of it divides 97. Here's what shortens it. Divisors come in pairs: if a times b equals n, then the smaller partner can never be bigger than the square root. So a sits at or below the root, and b at or above it.

Take 91. Its square root is about 9. 5, so you only test 2, 3, 5 and 7. Anything above 9 is already covered by a partner below it.

Quick tip: round the square root down, never up. And you can skip faster still. Skip evens above 2. Skip anything ending in 0 or 5.

Skip any number whose digits add to a multiple of 3. That's all. Here's the part that matters most. Take 25 — its square root is 5.

If nothing up to the square root of n divides n, nothing beyond it can. Let's test two numbers: 91 and 97. For each one, you only need the primes up to its square root. For 91, that's 2, 3, 5 and 7.

Two? 91 is odd, so no. Three? The digits add to ten, not a multiple of three.

Five? It ends in one. Seven? Seven times thirteen is 91 exactly, remainder zero.

A divisor exists, so 91 is composite — and we stop right there. Now 97, same four divisors. Two leaves a remainder of one. Three leaves one.

Five leaves two. Seven leaves six. Nothing divides it. The square root of 97 is about 9.

8, the list ends, and 97 is prime. Here's the point: one divisor is enough. Find one, and you never have to finish the list. Quick recap.

A prime has exactly two divisors: one and itself. Test the small primes. Stop at the square root of n. One hit means composite.

No hits means prime. And 91? It fails at 7. Seven times thirteen is ninety-one, so 91 is not prime.

Primes are the building blocks of every whole number.