The smallest common multiple

In the idli-vada game, the first 'idli-vada' comes at the smallest common multiple of the two numbers. For 3 and 5 it is 15. You do not need a formula: say both times tables from the start, and the first number that is in both is the smallest common multiple. For 2 and 5: the 2 table goes 2, 4, 6, 8, 10, and the 5 table goes 5, 10. They meet at 10.

A quicker way: write the multiples of the larger number in order, and stop at the first one that is also a multiple of the smaller. For 6 and 10: 10, 20, 30. Is 30 a multiple of 6? Yes, 6 × 5 = 30. So 6 and 10 first meet at 30.

Anshu and his friends play the game with two numbers that are both smaller than 10, and the first 'idli-vada' comes after 50. Which numbers could they be? Try 7 and 8: the multiples of 8 are 8, 16, 24, 32, 40, 48, 56, and 56 is the first that is also a multiple of 7 (7 × 8 = 56). So 7 and 8 first meet at 56, which is after 50.

The book also asks for the smallest number that is a multiple of all the numbers from 1 to 10 except 7. Such a number must be a multiple of 8, of 9 and of 5 (then 2, 3, 4, 6 and 10 come along for free). The common multiples of 8 and 9 are 72, 144, 216, 288, 360, ..., and the first one that is also a multiple of 5 is 360. Check: 360 is divisible by 1, 2, 3, 4, 5, 6, 8, 9 and 10.

Now include 7 as well. 360 is not a multiple of 7, and 7 has no factor in common with 360 other than 1, so we need 360 × 7 = 2520. The smallest number that is a multiple of all the numbers from 1 to 10 is 2520.