Question

The divisor and quotient of the number 6123 are same and the remainder is half the divisor. Find the divisor.

Answer 1

Let the divisor be x. Then, the quotient will be x and the remainder will be $\frac{x}{2}$
We know that Dividend = Quotient × Divisor + Remainder.
Thus, we get:
6123 = x × x + $\frac{x}{2}$
On multiplying both sides by 2, we get:
12246 = 2x² + x
$\Rightarrow$ 2x² + x – 12246 = 0
On splitting the middle term x as 157x – 156x, we get:
$\Rightarrow$ 2x² + 157x – 156x – 12246 = 0
$\Rightarrow$ x(2x + 157) – 78(2x + 157) = 0
$\Rightarrow$ (2x + 157)(x – 78) = 0
$\Rightarrow$ 2x + 157 = 0 or x – 78 = 0
$\Rightarrow$ x = –$\frac{157}{2}$ or x = 78
Since x is the positive divisor, x = 78.
Thus, the divisor is 78.