Question

A two digit is such that product of digits is 18 when 63 is subtracted from the number the digits interchange their places find the number

Answer 1

$\blue{\texttt{It is given that}}$,

$\blue{\texttt{product of digits }}$ = 18

$\blue{\texttt{lets suppose ten's digit}}$ = x

$\blue{\texttt{so, the unit digit }}$

= $\Large \frac{18}{x}$

Number = 10x +$\Large \frac{18}{x}$

we can get number $\blue{\texttt{by inter changing}}$,

= 10x + $\Large \frac{18}{x}$ +x

$\blue{\texttt{according to the question}}$,

10x +$\Large \frac{18}{x}$-$\Large \frac{180}{x}$-x =63

= 9x - $\Large \frac{162}{x}$=63

= 9 $x^{2}- 63x^{2}$ - 162 = 0

$x^{2}-7x -18 = 0$

$x^{2}-9x - 2x - 18 = 0$

$x (x - 9 ) +2 ( x- 9 ) =0$

$(x - 9 ) ( 2 + x) =0$

$x = 9 \:or x =-2$

$\blue{\texttt{now take x = 9}}$,

$\blue{\texttt{so , the number is }}$,

10x × 9 +$\Large \frac{18}{9}$

$\red{\textbf{ = 92}}$,

$\red{\textbf{hence, the number is 92}}$,

Answer 2

Let the two digit number be 10x + y

Now, xy = 18 --- (i) and,

(10x + y) - 63 = 10y + x ---- (ii)

form (ii) we have,

10x + y - 63 = 10y + x

=> 9x - 9y = 63

=> x - y = 7 --- (iii)

Now, (x - y)2 = (x + y)2 - 4xy

=> (x + y)2 = (x - y)2 + 4xy = (7)2 + 4(18) = 49 + 72 = 121

=> x + y = 11

so, x - y = 7 and x + y = 11

on adding we get,

2x = 18 => x = 9 => y = 2

Hence the two digit number is

10(9) + 2 = 92 Ans