Question

If three times the larger of the two numbers is divided by the smaller one, we get $4$ as quotient and $3$ as the remainder. Also, if seven times the smaller number is divided by the larger one, we get $5$ as quotient and $1$ as the remainder. Find the numbers.

Answer 1

Let the larger number be $x$ and the smaller one be $y$. We know that
Dividend $=$(Divisor $\times$ Quotient) $+$Remainder. (i)

When $3x$ is divided by $y$, we get $4$ as quotient and $3$ as remainder. Therefore, by using (i), we get

$3x=4y+3\Rightarrow 3x-4y-3=0$ (ii)

When $7y$ is divided by $x$, we get $5$ as quotient and $1$ as remainder. Therefore, by using (i), we get

$7y=5x+1\Rightarrow 5x-7y+1=0$ .(iii)

Solving equations (ii) and (iii), by cross-multiplication, we get

$\frac{x}{-4-21}=\frac{-y}{3+15}=\frac{1}{-21+20}$

$\frac{x}{-25}=-1,\frac{-y}{18}=-1$

$\Rightarrow x=25$ and $y=18$

Hence, the required numbers are $25$ and $18$.