Question

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

Answer 1

Hint: In this question remember to read the given information carefully where take divisor = d so that $remainder = \dfrac{d}{2}$ and also remember to use formula of dividend i.e. $Dividend = divisor \times quotient + remainder$, use this information to approach the solution.

Complete step-by-step answer:
According to the given information we have number 6123 whose divisor and quotient is same but remainder is the half of the divisor and we have to find the divisor
As we know that $Dividend = divisor \times quotient + remainder$
Let the divisor of the number 6123 be d
We know that divisor = quotient
Therefore, quotient = d
Also, we know that remainder is half of divisor
So, $remainder = \dfrac{d}{2}$
Substituting the given values in the formula i.e. $Dividend = divisor \times quotient + remainder$ we get
$6123 = d \times d + \dfrac{d}{2}$
$ \Rightarrow $$6123 = {d^2} + \dfrac{d}{2}$
And hence on taking the LCM and doing the cross multiplication, we got
$2{d^2} + d - 12246 = 0$
Now since this is a quadratic equation so we need to find the value of Discriminant and hence we have
We know that formula of Discriminant is given as; $D = {b^2} - 4ac$here a, b, and c are the coefficients of quadratic equation $a{x^2} + bx + c = 0$
Now substituting the value in the above formula, we get
$D = {\left( 1 \right)^2} - 4\left( 2 \right)\left( { - 12246} \right)$
$D = 1 + 8 \times 12246$
D = 97969
Now we know that quadratic formula to solve any quadratic equation is given as; $x = \dfrac{{ - b \pm \sqrt D }}{{2a}}$here x is the solution of the quadratic equation
Substituting the values in the above formula we get
$d = \dfrac{{ - b \pm \sqrt D }}{{2a}}$ and we know that$\sqrt D = \sqrt {97969} = 313$
And hence on putting the value we get
$d = \dfrac{{ - 1 \pm 313}}{{2 \times 2}}$
$ \Rightarrow $$d = \dfrac{{ - 1 \pm 313}}{4}$
And hence we have d = 78, - 78.5
Now on neglecting the negative value we got
d = 78
therefore, the divisor of number 6123 is 78.

Note: The trick behind the above question is to have the knowledge about the formula of dividend using which we will get a quadratic equation of degree two the solution of this equation can be found using the quadratic formula which is given as; $x = \dfrac{{ - b \pm \sqrt D }}{{2a}}$ and since the solution of the quadratic equation will be negative and positive so one should consider the only solution with positive value.