Question

A two digit number is such that the product of 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 $=10x+y$

It’s digits $x$ and $y$

Then, according to given question,

$x\times y=16$

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

Again, according to given question,

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

$10x+y-10y-x-54=0$

$9x-9y-54=0$

$x-y-6=0$

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

By equation (1) and (2) to, we get,

$x-y=6$

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

$x^2-16=6x$

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

$x^2-(8-2)x-16=0$

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

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

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

$x-8=0$ and $x+2=0$

$x=8$ and $x=-2$

Put the value of $x$ in equation $\left(1\right)$

$y=\frac{16}{x}$

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

But the number is $=10x+y$

$=10\times 8+2$

$=82$

Hence, this is the number.