Idli-vada: common multiples

Children sit in a circle and count 1, 2, 3, ... in turn. But there are three rules:

• On a multiple of 3 (3, 6, 9, 12, ...) you say 'idli' instead of the number.
• On a multiple of 5 (5, 10, 15, 20, ...) you say 'vada'.
• On a number that is a multiple of both 3 and 5, you say 'idli-vada'!

Anyone who makes a mistake is out. (It is easy to slip: at 10 many players say 'idli', but 10 is a multiple of 5, so it is 'vada'.)

A multiple of 3 is simply an answer in the 3 times table: 3, 6, 9, 12, 15, ... The first idli-vada number is 15, because 15 is in the 3 times table and in the 5 times table. Think of it as 15 = 3 × 5: both 3 and 5 are hidden inside 15.

Numbers that are multiples of two numbers at once are called common multiples. The common multiples of 3 and 5 are 15, 30, 45, 60, ...: exactly the multiples of 15. Picture two circles, one holding the multiples of 3 and one holding the multiples of 5. They overlap, and the common multiples sit in the shared middle. So the 10th time anyone says 'idli-vada' is at 10 × 15 = 150.

How often is each word said from 1 to 90?

• 'Idli' is said for every multiple of 3, counting the idli-vada ones too: 90 ÷ 3 = 30 times.
• 'Vada' is said for every multiple of 5, counting idli-vada too: 90 ÷ 5 = 18 times.
• 'Idli-vada' is said for the multiples of 15 up to 90: 15, 30, 45, 60, 75, 90, which is 6 times.

You can play with other pairs: 'idli' for multiples of the smaller number and 'vada' for multiples of the larger. Play with 4 and 8, and something strange happens: nobody ever says just 'vada'. Every multiple of 8 (8, 16, 24, ...) is also a multiple of 4, so each one is 'idli-vada'.