Question

If 460 is divided by a natural number, quotient is 6 more than five times the divisor and remainder is 1. Find quotient and diviser.

Answer 1

Let the divisor be x.
Dividend = 460
Quotient = 5x + 6
Remainder = 1
We know
$\mathrm{Divident}=\mathrm{Divisor}\times \mathrm{Quotient}+\mathrm{Remainder}$
$$\begin{gathered} 460=x\left(5x+6\right)+1 \\ \Rightarrow 460=5x^2+6x+1 \\ \Rightarrow 5x^2+6x-459=0 \\ \Rightarrow x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \Rightarrow x=\frac{-6\pm \sqrt{{\left(6\right)}^2-4\times 5\times \left(-459\right)}}{2\times 5} \\ \Rightarrow x=\frac{-6\pm \sqrt{36+9180}}{10} \\ \Rightarrow x=\frac{-6\pm \sqrt{9216}}{10} \end{gathered}$$
$$\begin{gathered} \Rightarrow x=\frac{-6\pm 96}{10} \\ \Rightarrow x=\frac{-6+96}{10},\frac{-6-96}{10} \\ \Rightarrow x=9,-10.2 \end{gathered}$$
Thus, divisor = 9, quotient = 5x + 6 = $5\times 9+6=45+6=51$