Question

A two digit number is 4 times the sum and 3 times the product of its digits .find the number.

Answer 1

Let the digits be x and y
Therefore the number is 10x + y
According to question

Case 1
4(x +y) = 10x +y
4x + 4y = 10x + y
0 = 10x - 4x - 4y +y
0= 6x -3y
0 = 2x - y
y = 2x ...... (1)

Case 2
3 xy = 10x + y
3x (2x) = 10x + 2x. (From equation (1))
6x^2 = 12x
$\frac{ {x}^{2} }{x} = \frac{12}{6}$

x = 2
&
Y = 2x = 2 × 2 = 4

Ans. The number is 24 or 42

Answer 2

A two digit positive integer can be expressed as 10x+y, where x is the tens digit and y is the ones digit.  For example, 37 = 3(10)+7.

 

So, 10x+y = 4(x+y)  and 10x+y = 3xy

 

10x+y = 4x+4y

 

      6x = 3y

 

        y = 2x

 

Since y = 2x and 10x+y = 3xy, we have 10x+2x=3x(2x)

 

                                                                12x = 6x2             6x2-12x = 0

 

                                                                                       6x(x-2) = 0   x = 0 or x = 2

 

                                                                                        Since x can't be zero, x = 2

 

                                                                                                                        y = 2x = 4

 

The number is 24.