Question

On dividing$624$and$2121$by a $3$-digit number, we get the same remainder. Find that number?

Answer 1

By division algorithm, we know that -

$\text{Dividend = (divisor×quotient)+remainder}$

Let the three-digit number be $\text{N}$, the remainder in each case be$\text{"r"}$, quotient when$624$ divided by$\text{N}$ be $\text{"m"}$ and quotient when divided$2121$divided by $\text{N}$be $\text{"n"}$.

Using the division algorithm,

$$\begin{gathered} \text{624 = Nm+r ------------------}\left(i\right) \\ 2121=Nn+r------------------\left(ii\right) \end{gathered}$$

Subtracting equation (i) from (ii) we wiil get -

$$\begin{gathered} \text{1497 = N(n-m)} \\ 3\text{×499 = N(n-m)} \end{gathered}$$

Since, a three-digit number is $499$, therefore the value of $\text{N}$ is $499$.

Hence, the three-digit number is $499$.