Question

A number when divided by $136$ , leaves $36$ as remainder. If the same number is divided by $17$ , what will be the remainder?
(A) $0$
(B) $1$
(C) $2$
(D) $3$

Answer 1

Hint:
Analyse the question properly and with given information establish a relationship using ${\text{Dividend = }}\left( {{\text{Divisor}} \times {\text{Quotient}}} \right){\text{ + Remainder}}$ . You can assume some arbitrary variables for the unknowns. Notice that $136 = 17 \times 8$ and use it in the above equation and then change $36$ as $36 = 17 \times 2 + 2$. Now rearrange them to form a comparable equation to find the remainder.

Complete step by step solution:
Here in this problem, a number is divided by $136$ and it leaves a remainder of $36$ . Then again that same number is divided by $17$ and we have to calculate the remainder for this division.
As we know that when an operation of division takes place, then we have a dividend equal to the sum of the remainder and product of quotient and divisor. And the dividend will be greater than the divisor for getting quotients more than one. And the value of the remainder will lie between zero and divisor.
$ \Rightarrow {\text{Dividend = }}\left( {{\text{Divisor}} \times {\text{Quotient}}} \right){\text{ + Remainder}}$
According to the given operation in the question, let’s assume that the dividend number is $'x'$ and the quotient is any whole number $'n'$ . Form the above relation, we can write:
$ \Rightarrow x = 136 \times n + 36$ ……..(i)
Now, the same dividend is again divided by divisor $17$ and this can be represented as:
$ \Rightarrow x = 17 \times m + r$ ………..(ii)
Here $'m'$ is assumed to be the quotient and $'r'$ is the remainder.
We can also express $136$ as the multiple of $17$ , that is:
$ \Rightarrow 136 = 17 \times 8$
Let’s substitute this value in the equation (i), we will get:
$ \Rightarrow x = 136 \times n + 36 = \left( {17 \times 8} \right) \times n + 36$
Also, the remainder $36$ can be expressed as $36 = 17 \times 2 + 2$ . Putting this back into the above expression will give us:
$ \Rightarrow x = \left( {17 \times 8} \right) \times n + 36 = \left( {17 \times 8} \right)n + \left( {17 \times 2 + 2} \right) = 17 \times 8n + 17 \times 2 + 2$
Now we can transform the above expression in form of relation (ii), we get:
$ \Rightarrow x = 17 \times \left( {8n + 2} \right) + 2$
On comparing the above equation with relation (ii) we have, $(8n + 2)$ is the quotient same as $'m'$ and the remainder $'r'$ will have the value $2$ .
Therefore, the remainder when the same number is divided by $17$ will be \[2\] .

Hence, the option (C) is the correct answer.

Note:
Remember that for questions like this, we always take quotient as a whole number. So when we compared $(8n + 2)$ will $'m'$ they were both whole numbers. An alternate approach for the same can be to use $136 = 17 \times 8$ in equation (i) and transform it into $x = 17 \times \left( {8n + 2} \right) + 2$ and then use relation ${\text{Dividend = }}\left( {{\text{Divisor}} \times {\text{Quotient}}} \right){\text{ + Remainder}}$ to determine the remainder of the operation.