Primes, Composites, Square and Triangular Numbers

A prime number has exactly two factors: one and itself. Seven can only be arranged as a single row of seven dots — there is no rectangle — so seven is prime. Twelve makes rectangles: two by six, three by four. It has six factors, so it is composite. Two is the only even prime, because every other even number has two as a factor. And one is neither prime nor composite: it has only one factor, itself, and the definition asks for two. That is why the list of primes starts at two.

To test whether a number is prime, try dividing by the primes in order. Ninety-one: it is odd, so not two. Its digits add to ten, so not three. It does not end in zero or five, so not five. Seven? Seven times thirteen is ninety-one. So ninety-one is composite, and it catches people because seven times thirteen is outside the tables they know. Two rules make the test fast. Use the divisibility tests for two, three and five. And stop once the prime you are trying, multiplied by itself, is bigger than the number: for ninety-seven, after seven fails you can stop, because eleven times eleven is already one hundred and twenty-one.

A square number is a number of dots that makes a square: one, four, nine, sixteen, twenty-five. Each is a number multiplied by itself. Two things are worth knowing about them. The gaps between consecutive squares are the odd numbers — plus three, plus five, plus seven — which lets you extend the list without multiplying. And a square number has an odd number of factors, because one of its factor pairs is a number with itself: thirty-six is six times six, and that six counts once. Learn the squares up to twelve times twelve; questions that ask you to spot one give you no time to work it out.

A triangular number is a number of dots that makes a staircase: one, three, six, ten, fifteen. Each is the previous one plus the next counting number — plus two, plus three, plus four. So the sixth triangular number is one plus two plus three plus four plus five plus six, twenty-one. They turn up as handshakes in a room, matches in a tournament, the pins in ten-pin bowling. The one to notice: thirty-six is both a square number, six times six, and a triangular number, one to eight added up. If a question asks for a number that is both, thirty-six is almost always it.

Test questions combine these by asking you to list and then count or calculate. Between one and twenty there are eight primes, four square numbers and five triangular numbers. The sum of the first five primes is two plus three plus five plus seven plus eleven, twenty-eight. The product of the first three is two times three times five, thirty. The smallest prime greater than ninety takes a walk: ninety-one is seven times thirteen, ninety-three is three times thirty-one, and the first that survives every test is ninety-seven. Write the list first; every one of these is easy once the list is on the page. Quiz and exercises now.