A two digit number is obtained by multiplying the sum of the digits by 8. Also, it is obtained by multiplying the difference of the digits by 14 and adding 2. Find the number.

Let x be the digit at ten's place and y be the digit at unit's place.

Number = 10(x) + y = 10x + y

Given,

Number is obtained by multiplying the sum of the digits by 8.

∴ 8(x + y) = 10x + y

⇒ 8x + 8y = 10x + y

⇒ 10x - 8x = 8y - y

⇒ 2x = 7y

⇒ x = $\dfrac{7}{2}$y ……….(1)

Number is obtained by multiplying the difference of the digits by 14 and adding 2.

⇒ 14(x - y) + 2 = 10x + y ………….(2)

or,

⇒ 14(y - x) + 2 = 10x + y ………….(3)

Solving equation (2),

⇒ 14x - 14y + 2 = 10x + y

⇒ 14x - 10x = y + 14y - 2

⇒ 4x = 15y - 2 ………..(4)

Substituting value of x from equation (1) in equation (4), we get :

$\Rightarrow 4 \times \dfrac{7}{2}y = 15y - 2 \\[1em] \Rightarrow 14y = 15y - 2 \\[1em] \Rightarrow 15y - 14y = 2 \\[1em] \Rightarrow y = 2.$

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

$\Rightarrow x = \dfrac{7}{2}y = \dfrac{7}{2} \times 2$ = 7.

Number = 10x + y = 10(7) + 2 = 70 + 2 = 72.

Solving equation (3),

⇒ 14(y - x) + 2 = 10x + y

⇒ 14y - 14x + 2 = 10x + y

⇒ 14y - y = 10x + 14x - 2

⇒ 13y = 24x + 2 …………(5)

Substituting value of x from equation (1) in equation (5), we get :

$\Rightarrow 13y = 24 \times \dfrac{7}{2}y + 2 \\[1em] \Rightarrow 13y = 84y + 2 \\[1em] \Rightarrow 84y - 13y = -2 \\[1em] \Rightarrow 71y = -2 \\[1em] \Rightarrow y = -\dfrac{2}{71}.$

This is not possible as y cannot be a fraction.

Hence, number = 72.