Question

[IMC 1997 Q12] Gill is nine years old this year so she was fascinated to discover that an integer is divisible by 9 precisely when the sum of its digits is also divisible by 9. If the six-digit number “1*3456” is to be a multiple of 9, what must the missing digit be?

Answer 1

If it must be divisible by 9 the sum of digits should we divisible by 8

We have 13456

= 19

We cannot reduce a digit so the nearest next divisible by 9 is 27 which is 8 numbers away from 19

So the missing digit should be 8.

If we take numbers more than 27 it will be a 2 digit so it won't fit.

Answer= 8