Question

A two-digit number is four times the sum of its digits and twice the product of the digits. Find the number.

Answer 1

Let the tens digit be x and the units digit be y.Then the two-digit number is (10x + y).
Given: Two-digit number = 4 times sum of its digits
10x + y = 4(x + y)
⇒ 10x + y = 4x + 4y
⇒ 6x = 3y
x = y/2.... (1)
Given: Two-digit number = Twice the product of its digits
⇒ 10x + y = 2(xy)
⇒10x + y – 2xy = 0 …(2) 
substituting x= y/2 from equation (1) and equation (2) we get
10 * y/ 2 + y - 2 * y / 2* y = 0
⇒5y + y – y2 = 0
⇒y2 – 6y = 0
⇒y (y – 6) = 0
⇒ y = 0 or y – 6 = 0
We reject y = 0 because then x would also be zero.
Therefore, y = 6
Substituting this value of y in equation 1, we get
x = 6/ 2 = 3
Hence required number is 
(10 × 3 + 6) = 36