Question

A two digit number is such that the product of its digits is 8. When 18 is added to the number they interchange their places. Determine the number?
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Answer 1

Let the units digit be x and the tens digit be $y$.
Then the value of the number is $10y+x$
We have $xy=8$..... .(1)
When $18$ is added to the number, the digits interchange their places.
So, we will have $y$ in units place and $x$ in tens place.
The new value of number is $10x+y$.
We can write this as the equation: $10y+x+18=10x+y$
$\Rightarrow 10x-x+y-10y=18$
$\Rightarrow 9x-9y=18$
$\Rightarrow x-y=2$........(2)
Square both sides to get $x^2-2xy+y^2=4$....... (3)
From (1) , $4xy=32$....... (4)
Add (3) and (4) to get $x^2+2xy+y^2=36$
$\Rightarrow (x+y{)}^2=36$
Taking square root we get:$x+y=6$....... (5)
Adding (2) and (5) we get $2x=8$ or $x=4$
Substituting $x=2$ in (2), we get $4-y=2$ or $y=2$
So, the original number is $24$.