Question

A two digit number is such that the product of the digits is $16$. When 54 is subtracted from the number, the digits interchange their places. Find the number.

Answer 1

Let the two digit number is $10x+y$

According to given question,

$x\times y=16$

$y=\frac{16}{x}.......\left(1\right)$

Now again,

$10y+x=10x+y-54$

$\Rightarrow 10y+x-10x-y=-54$

$\Rightarrow 9y-9x=-54$

$\Rightarrow 9(y-x)=-54$

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

put the value of y by equation $\left(1\right)$ in equation $\left(2\right)$

$x-\frac{16}{x}=6$

$\Rightarrow x^2-16=6x$

$\Rightarrow x^2-6x-16=0$

$\Rightarrow x^2-8x+2x-16=0$

$\Rightarrow x(x-8)+2(x-8)=0$

$\Rightarrow (x-8)(x+2)=0$

$\Rightarrow x-8=0x+2=0$

$\Rightarrow x=8x=-2$

Put the value of x in equation (1) and we get,

$y=\frac{16}{2}=8$

Then, the number is

$10x+y$

$=10\times 2+8$

$=28$

Hence, this is the answer.