Here is a big number: 3,540. Without doing any dividing at all, can you already tell whether it splits evenly into 2, into 5, and into 10? Take a good look at it before anyone says a word. What is it about the number that gives the game away?

Four hundred squares show the shading for one divisibility rule at a time. Look where the shading falls, and see which tests only need the last digit and which need something more.

Now we work through some divisors together on the grid: first we shade the multiples of 2, then 4, then 6. When we shade the multiples of two, every second cell lights up, because two goes into every even number. There will be fewer cells for four, and we will check each one with the last-two-digits test. Six is the interesting one. A multiple of six has to pass two tests at once, because 6 = 2 × 3: the number must be even (the 2 test) and its digit-sum must divide by 3 (the 3 test).