Question

Divide 53068 by 257 and check the result by the division algorithm.

Answer 1

Hint: We solve this question by first dividing the dividend which is 53068 by the divisor 257 using the long division method. Then we get the required quotient and the remainder. Then we shall verify this using the division algorithm which is given as $Dividend=Divisor\times q+r,$ where q is the quotient and r is the remainder obtained after division. We use these values to verify the result.

Complete step-by-step solution:
In order to solve this question, let us first divide the two terms using the long division method represented as follows,
$\Rightarrow 257\overline{\left){53068}\right.}$
We now take the first 3 digits of the dividend which is 530. We need a number which is a multiple of 257 which is just less than 530. We get 514 as the closest number by multiplying 257 by 2. This is represented as
$\Rightarrow 257\overset{2}{\overline{\left){\begin{align}
  & \text{ }53068 \\
 & -514 \\
 & \overline{\text{ }16} \\
\end{align}}\right.}}$
Next, we bring down the next digit from the dividend and try to divide,
$\Rightarrow 257\overset{2}{\overline{\left){\begin{align}
  & \text{ }53068 \\
 & -514 \\
 & \overline{\text{ }166} \\
\end{align}}\right.}}$
But we can see that 257 cannot divide 166 as it is a smaller number. Hence, we need to bring down another digit from the dividend and to do so, we need to add a 0 in the quotient.
$\Rightarrow 257\overset{20}{\overline{\left){\begin{align}
  & \text{ }53068 \\
 & -514 \\
 & \overline{\text{ }1668} \\
\end{align}}\right.}}$
Next, we 257 by 6 to get 1542 which is the closest number to 1668 which is a multiple of 257. We subtract this from 1668 to get the remainder.
$\Rightarrow 257\overset{206}{\overline{\left){\begin{align}
  & \text{ }53068 \\
 & -514 \\
 & \overline{\begin{align}
  & \text{ }1668 \\
 & \text{ }-1542 \\
 & \overline{\text{ }126} \\
\end{align}} \\
\end{align}}\right.}}$
This 126 cannot be divided any further by 257, hence the quotient obtained is 206 and the remainder obtained is 126.
Now we need to verify this. In order to do so, we use the formula given by
$\Rightarrow Dividend=Divisor\times q+r$
We have the required values for dividend which is 53068, divisor as 257, quotient as 206 and remainder as 126. Substituting these values, we need to check if both the sides of the equation are the same. If it is then it is verified.
$\Rightarrow 53068=257\times 206+126$
Multiplying the terms on the right-hand side,
$\Rightarrow 53068=52942+126$
Adding the two terms on the right-hand side,
$\Rightarrow 53068=53068$
Since the left-hand side is equal to the right-hand side the division is verified.

Note: We need to be careful while dividing to make sure we multiply the divisor by a number such that it is less than the current remainder. If it is more, then we will get a wrong answer. We also need to note that if the current remainder is not divisible, we bring down an extra digit down from the dividend by adding an extra 0 in the quotient.