Dear Uncle Colin,

At a Mathscounts competition, a contestant was asked “how many six-digit positive integers are divisible by 1000 but not by 400?”. Within four seconds, they correctly answered 450 – how on earth could they do that so quickly?

- Mathematical Evaluations Normally Take A Lot Longer, Yes?

Hi, MENTALLY, and thanks for your message!

I think the short answer is ‘practice’, but this actually falls out quite quickly with a little thought - and presumably Mathscounts competitors have plenty of experience with this sort of question.

The six-digit numbers go from 100,000 to 999,999, and 900 of them are divisible by 1000 - starting at 100,000 and ending at 999,000. (It’s easy to make an off-by-one error here if you’re not careful.)

All 1000-numbers are divisible by 200, but only half of them are divisible by 400 (the ones with an even digit before the comma.) They alternate, so half of the 900 candidate numbers are not multiples of 400, giving a total of 450.

Alternatively, you can divide the problem by 100, in a sense: how many four-digit numbers are multiples of 10 but not multiples of 4? The numbers work out the same.

Hope that helps!

- Uncle Colin