Question

A two digit number is 4 times the sum of its digits and twice the product of its digits.The number is ------

Answer 1

Let the digit in the ones place be $x$ and tens place be $y$

Hence the two digit number $=10y+x$

Given that the two digit number $=4$ times sum of its digits

According to the question,

$\Rightarrow$ $10y+x=4(x+y)$

$\Rightarrow$ $10y+x=4x+4y$

$\Rightarrow$ $3x-6y=0$

$\Rightarrow$ $x=2y$ ------- ( 1 )

It is also given that the two digit number $=2$ times product of its digits

$\Rightarrow$ $10y+x=2xy$

Divide by $xy$ both the sides, we get

$\Rightarrow$ $\frac{10}{x}+\frac{1}{y}=2$

Put $x=2y$ from (1), we get

$\Rightarrow$ $\frac{10}{2y}+\frac{1}{y}=2$

$\Rightarrow$ $\frac{5}{y}+\frac{1}{y}=2$

$\Rightarrow$ $\frac{6}{y}=2$

$\therefore$ $y=3$

$\Rightarrow$ $x=2y=2\left(3\right)=6$

$\Rightarrow$ The two digit number $=(10y+x)=10\left(3\right)+6=36$