Question

In a two digit no. the digit at unit place is equal the square of the digits at tens place. y 54 is added to the no. the digits get interchanges. Find the no.

Answer 1

Suppose that,

The two digit number is $10x+y$

Then, the unit palace is $y$ and tens palace is $x$

According to given question,

$y=x^2$…… (1)

Then,

Again, according to given question,

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

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

$\Rightarrow 9x-9y+54=0$

$\Rightarrow x-y+6=0......\left(2\right)$

From equation (1) to,

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

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

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

$\Rightarrow x^2-(3-2)x-6=0$

$\Rightarrow x^2-3x+2x-6=0$

$\Rightarrow x(x-3)+2(x-3)=0$

$\Rightarrow (x-3)(x+2)=0$

For,

$x-3=0$

$x=3$

Put in (2)

Then,

$x-y+6=0$

$3-y+6=0$

$-y+9=0$

$y=9$

Then,

The number is

$10x+y$

$\Rightarrow 10\times 3+9$

$\Rightarrow 39$

Hence, this is the answer.