Question

If $21y5$ is a multiple of $9$, where $y$ is a digit, then find the value of $y$.

Answer 1

Hint: To solve this problem, we apply the divisibility rule of 9. The divisibility rule of 9 says that if the sum of all the digits of a number is divisible by 9, then the entire number is divisible by 9.

We find the sum of the digits of $21y5$ which will be divisible by 9. Here $y$ can take the values of $0 - 9$, as it is a single digit.

We try to make an equation after finding the multiple of 9. Solving the linear equation, we find the possible value of y.

Complete step by step answer:

Given $21y5$ is a multiple of 9. This means  $21y5$ is divisible by 9.

So, the sum of the digits of $21y5$ has to be divisible by 9.

This means $2+1+y+5$

$=y+8$ is divisible by 9.

$y$ is a single digit. 

We need a number for $y$ to make $y+8$ is a multiple of $9$.

Now we find the multiple of 9 greater than 8. 

The first three multiples of $9$ are 9, 18, 27.

If the sum is 9 then the equation becomes $y+8=9$ which gives $y=1$.

Now if the sum is 18 then the equation becomes $y+8=18$ which gives $y=10$. This is not possible as y is a single digit number.

We don’t need to proceed further.

Therefore, the value of $y$ is 1.

Note: We need to keep in mind that y is only a single digit number. So, the range of y is $y\in \left[ 0,9 \right]$. When we found that after 1 the next possible term is 10, we considered not to proceed further as the term will keep increasing.