Question

Product of digits of a two-digit number is $18$. If $27$ is subtracted from the number then digits interchanged mutually. Find the number.

Answer 1

Let the two digit of the number be $a,b$ at the tenth and hundred place respectively,

therefore the product of two digits be

$\Rightarrow ab=18-\left(1\right)$

it is given that,

$\Rightarrow$$10a+b-27=10b+a$ -(2)

solving (2) we get

$\Rightarrow$$a-b=3$ -(3)

now using the formula

$\Rightarrow$${(a+b)}^2$=${(a-b)}^2+4ab$

using this

$\Rightarrow$${(a+b)}^2$=${\left(3\right)}^2+4\ast 1\ast 18$

we get,

$\Rightarrow$$(a+b)=9.....\left(5\right)or(a+b)=-9.......\left(6\right)$

solving eq (3 )and (5), (3) and (6)

we get

$\Rightarrow a=6,3$

and

$\Rightarrow b=-3,-6$

Therefore the number could be $63$ or $-36$.