Question

A two digit number is four times the sum of the digits. It is also equal to 3 times the product of digits. Find the number ?

Answer 1

Let the digits of the required number be x and y.
Now, the required number is 10x + y.
According to the question,
10x + y = 4(x + y)                
So,
6x − 3y = 0

\[\Rightarrow\]2x − y = 0

\[x = \frac{y}{2}\]                                               .....(1)

Also, 
10x + y = 3xy                                            .....(2)
From (1) and (2), we get

\[10\left( \frac{y}{2} \right) + y = 3\left( \frac{y}{2} \right)y\]
\[ \Rightarrow 5y + y = \frac{3}{2} y^2 \]
\[ \Rightarrow 6y = \frac{3}{2} y^2 \]

\[\Rightarrow y^2 - 4y = 0\]
\[ \Rightarrow y(y - 4) = 0\]
\[ \Rightarrow y = 0, 4\]

So, x = 0 for y = 0 and x = 2 for y = 4.

Hence, the required number is 24.