Question

What least number should be replaced for $ * $ so that the number $67301 * 2$ is exactly divisible by $9$?

Answer 1

Hint: If a number is divisible by nine, then the sum of its digits must also be divisible by nine. So we have to find the answer such that the sum of digits is a multiple of nine. Since $ * $ is a number less than or equal to nine, we get the answer easily.

Complete step-by-step solution:
Given the number $67301 * 2$.
We have to find the number $ * $ so that $67301 * 2$ is exactly divisible by nine.
If a number is divisible by nine, then the sum of its digits must also be divisible by nine.
So we have to find $ * $ such that, $6 + 7 + 3 + 0 + 1 + * + 2$ is a multiple of nine.
That is, $19 + * $ is a multiple of nine.
We know the least multiple of nine greater than or equal to $19$ is $27$.
So we can write, $19 + * = 27$
This gives, $ * = 27 - 19 = 8$
So the least number that can be replaced by $ * $ so that $67301 * 2$ is exactly divisible by nine is $8$.

$\therefore $ The answer is $8$.

Note: Certain numbers like nine have divisibility rules. So we can find the missing digits using them. But this may not work all time. So we need to use different methods.
In the question, it is asked for the least number satisfying the condition. If it could be a two digit number we have more options like $17,26...$
Since $19 + 17 = 36$ and $19 + 26 = 45$ and these all are multiples of nine, these numbers would have worked.
That is the importance of the word ‘least’ here.