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?

You are about to see the same grid of numbers four times over, shaded for one divisibility rule at a time. Watch where the shading falls each time, and work out which tests you could pass with only the last digit, and which ones ask you for more than that.

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).

Now we shade the numbers that pass each test in turn: first every multiple of 3, then every multiple of 6 (a number that passes both the 2 test and the 3 test), then every multiple of 9, and finally every multiple of 12. A multiple of 12 is a good one to finish on, because every one of them passes both the 3 test and the 4 test at the same time. Each round asks a little more of the digit-sum rule than the last.