If three times the larger of the two numbers is divided by the smaller, then the quotient is 4 and remainder is 5. If 6 times the smaller is divided by the larger, the quotient is 4 and the remainder is 2. Find the numbers.

Let two numbers be x and y, where x > y.

Given,

Three times the larger divided by the smaller gives quotient 4.

⇒ 3x = 4y + 5

⇒ x = $\dfrac{4y + 5}{3}$     …..(1)

Given,

Six times the smaller divided by the larger gives quotient 4 and remainder 2.

⇒ 6y = 4x + 2     …..(2)

Substituting value of x from equation (1) in (2), we get :

$\Rightarrow 6y = 4 \Big(\dfrac{4y + 5}{3}\Big) + 2 \\[1em] \Rightarrow 6y = \Big(\dfrac{16y + 20}{3}\Big) + 2 \\[1em] \Rightarrow 6y = \Big(\dfrac{16y + 20 + 6}{3}\Big) \\[1em] \Rightarrow 6y = \Big(\dfrac{16y + 26}{3}\Big) \\[1em] \Rightarrow 6y \times 3 = 16y + 26 \\[1em] \Rightarrow 18y = 16y + 26 \\[1em] \Rightarrow 18y - 16y = 26 \\[1em] \Rightarrow 2y = 26 \\[1em] \Rightarrow y = \dfrac{26}{2} \\[1em] \Rightarrow y = 13.$

Substituting value of y in equation (1), we get :

$\Rightarrow x = \dfrac{4y + 5}{3} \\[1em] \Rightarrow x = \dfrac{4 \times 13 + 5}{3} \\[1em] \Rightarrow x = \dfrac{52 + 5}{3} \\[1em] \Rightarrow x = \dfrac{57}{3} = 19.$

Hence, the numbers are 19 and 13.