Question

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

Answer 1

Answer:

24

Step-by-step explanation:

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.

Answer 2

Suppose unit digit be p

Tens digit be m

Number = 10m + p

Now

10m + p = 4(p + m)

6m - 3p = 0

2m - p = 0 .......... (1)

Also

10m + p = 3pm

10m + 2m = 3pm ..... (From equation 1)

12m = 3pm

3m (4 - p) = 0

m = 0

p = 4

Avoid 0

p = 4

From equation 1 we get :-

2m - 4 = 0

2m = 4

${\implies m = \dfrac{4}{2}}$

Required Number :-

= 2 × 10 + 4

= 24

Q.E.D