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 divisor.

Answer 1

Let the divisor be x. 
Dividend = 460
Quotient = 5x + 6
Remainder = 1
We know 

\[\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}\] 

\[460 = x\left( 5x + 6 \right) + 1\]
\[ \Rightarrow 460 = 5 x^2 + 6x + 1\]
\[ \Rightarrow 5 x^2 + 6x - 459 = 0\]
\[ \Rightarrow x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a}\]
\[ \Rightarrow x = \frac{- 6 \pm \sqrt{\left( 6 \right)^2 - 4 \times 5 \times \left( - 459 \right)}}{2 \times 5}\]
\[ \Rightarrow x = \frac{- 6 \pm \sqrt{36 + 9180}}{10}\]
\[ \Rightarrow x = \frac{- 6 \pm \sqrt{9216}}{10}\] 

\[\Rightarrow x = \frac{- 6 \pm 96}{10}\]
\[ \Rightarrow x = \frac{- 6 + 96}{10}, \frac{- 6 - 96}{10}\]
\[ \Rightarrow x = 9, - 10 . 2\] 

Thus, divisor = 9, quotient = 5x + 6 = \[5 \times 9 + 6 = 45 + 6 = 51\]