Every Number's Secret Ingredient List Prime Time

In a circle, children count upward, calling 'idli' on every multiple of 3, 'vada' on every multiple of 5, and 'idli-vada' whenever a number is a multiple of both. Playing this out reveals the numbers that get double-named -- 15, 30, 45, and so on -- are exactly the common multiples of 3 and 5. Playing the same game with different starting pairs, like 4 and 6, changes which numbers get the double name, but the underlying idea stays fixed: a common multiple is simply a number appearing on both individual multiples lists at once.

Grumpy hides a treasure at some number on a long number strip, and Jumpy tries to reach it by choosing a jump size and hopping that same distance every time, starting from 0. Jumpy succeeds exactly when the chosen jump size is a factor of the treasure's position -- a number that divides it exactly with nothing left over. When two treasures are hidden at once, only a jump size that is a factor of BOTH numbers can land on both, making it a common factor of the two. This single idea, tried across many treasure pairs, builds toward the definition used throughout the rest of the chapter: the factors of a number are exactly the jump sizes that land on it exactly.

Arranging a given number of figs into a neat rectangular array (rows times columns) works for some counts in several different ways and for others in only one way at all. A count like 12 can be arranged as 1x12, 2x6, or 3x4, but a count like 7 can only ever be arranged as 1x7 -- no other rectangle fits. Numbers like 7, with exactly two factors (1 and itself), are called prime; numbers with more than two factors are composite. Around 2200 years ago, the Greek mathematician Eratosthenes devised a systematic way to find every prime up to any limit: write out every number, then repeatedly cross out every multiple of each remaining unfound number, starting from 2 -- whatever survives uncrossed is prime. This method, the Sieve of Eratosthenes, still works exactly the same way today.

Extending the treasure game with a new rule -- no jump of size 1 allowed -- reveals which number pairs are impossible to guard against every possible jump: numbers that share no common factor larger than 1. Such pairs are called co-prime, meaning their only common factor is 1, even if neither number is itself prime (8 and 15 are co-prime, though neither is a prime number). A related craft activity threads string between evenly spaced pegs on a circle, skipping a fixed gap each time; the resulting pattern closes into one single unbroken loop exactly when the peg count and the gap size are co-prime, and instead breaks into several smaller separate loops whenever they share a common factor.

Checking whether 56 and 63 are co-prime by picking just one convenient factor pair each -- 56=14x4 and 63=21x3 -- can mislead: no factor looks shared at first glance, suggesting co-prime. But breaking both numbers down completely into prime factors instead, 56=2x2x2x7 and 63=3x3x7, reveals a shared prime, 7, meaning they are NOT co-prime after all. A single convenient factor pair can hide a shared factor that only becomes visible once every number is broken down the whole way to primes. This complete breakdown, called prime factorisation, always ends at the exact same set of primes no matter which factor is peeled off first: 36 reached via 4x9, or via 2x18, or via 6x6, or via 3x12, always bottoms out at 2x2x3x3. This uniqueness is precisely what makes prime factorisation reliable for testing co-primality and divisibility, where a single lucky-looking factor pair cannot be trusted.

Some divisibility can be spotted directly from a number's last one or two digits, without factorising at all. A number is divisible by 10 exactly when its units digit is 0; by 5 when its units digit is 0 or 5; by 2 when its units digit is even. Divisibility by 4 needs the last TWO digits: if the two-digit number they form is itself divisible by 4, so is the whole number (528 is divisible by 4 because 28 is; 530 is not, because 30 is not). Divisibility by 8 extends the same idea to the last THREE digits. Rules for 3, 6, 7, and 9 need a different kind of test entirely, one that looks at the sum of all the digits rather than just the last few -- and this chapter deliberately leaves those for a later class.