Question

For what possible values of $x$ the number $36x$ i.e. $3\times 100+6\times 10+x$ is divisible by $3$ ?

Answer 1

Hint: In this question we have been given a digit whose unit term is unknown. We have to find the value of the unit digit such that the number is divisible by $3$.By the divisibility test of $3$ we know that a number is divisible by $3$ only if the sum of its digit is divisible by $3$ using this concept we will find out the value of the unknown digit and get our desired answer.

Complete step by step answer:
The number is given as follows: $36x$
Where it is denoted as $3\times 100+6\times 10+x$.
Now we have to find a value of $x$ such that the above number is divisible by $3$.
By divisibility test of $3$ sum of all the digits of the number should be divisible by $3$.
So sum of all the digit of the given number is as follows: $3+6+x$
$\Rightarrow 9+x$……$\left( 1 \right)$
Now as we can see that $9$ is divisible by $3$ so one value of $x$ can be $0$.
$\Rightarrow \left( 9+0 \right)\div 3=3$
Next $x$ should be a multiple of $3$ so that the equation (1) is satisfied. So $x=3,6,9$ and we can see that the equation (1) is satisfying for all the three values as follows:
$\Rightarrow \left( 9+3 \right)\div 3=4$
$\Rightarrow \left( 9+6 \right)\div 3=5$
And lastly
$\therefore \left( 9+9 \right)\div 3=6$

Hence possible value for $x$ such that the number $36x$ is divisible by $3$ is $0,3,6,9$.

Note: Divisibility test or rule is used to make division procedure easy and fast. If we know the divisibility rule for all the numbers, no matter how big the division problem is, it becomes quite easy to solve. Divisibility tests of numbers $2,3,4,5,6,7,8,9,10$ are quite easy to understand. But divisibility tests of $11,13$ are complex and take time to understand.