Question

what is the missing digit which makes the number $47\_12$ exactly divisible by 9.
A. 1
B. 2
C. 3
D. 4

Answer 1

Hint: We first state the division rule of 9 which is the sum of the digits in that number has to be divisible by 9. Then we assume the missing number and find the sum. We find the closest possible number divisible by 9 of that sum and find the missing number from the linear equation.

Complete step by step answer:
Here we have to find a missing digit in $47\_12$ such that it’s exactly divisible by 9.
We are going to use the division rule of 9.
The rule says that a number is exactly divisible by 9 if the sum of the digits in that number is divisible by 9.
For example: if $\left( abc \right)$ is the number with a, b, c being the hundredth, tenth and unit placed numbers respectively then the number is divisible by 9 if $\left( a+b+c \right)$ is divisible by 9. We take 4875. We find the sum of the digits which is $\left( 4+8+7+5 \right)=24$ which is not divisible by 9. It means 4875 is not divisible by 9. $\dfrac{4875}{9}=541.6$.
Now for our given number $47\_12$, let’s assume that the missing number is x.
The sum of the digits will be $\left( 4+7+x+1+2 \right)=14+x$.
For $47\_12$ to be divisible by 9, $14+x$ has to be divisible by 9.
Now the value of x is positive. So, the value of $14+x$ will be greater than 14. Also, it has to be divisible by 9.
Nearest number of 14 divisible by 9 is 18.
So, $14+x=18\Rightarrow x=18-14=4$.
The missing digit is 4.

So, the correct answer is “Option D”.

Note: We could have gone to the next possible multiple of 9 after 18 which is 27. But in that case the value of x would have been $14+x=27\Rightarrow x=27-14=13$, which is not possible as the digit is of only 1 digit. It can’t be two digits. So, the only possible option for the sum is 18.