Question

The units digit of a two-digit number is $2$ more than the tens digit. If the number is subtracted from the sum of the squares of its digits the result is two-thirds of the product of the digits. What is the number?

• A. $62$
• B. $58$
• C. $68$
• D. $86$

Answer 1

The correct option is C $68$

given the units digit of a two-digit number is $2$ more than the tens digit. and the number is subtracted from the sum of the squares of its digits the result is two-thirds of the product of the digits.

Let the tens digit be x.x

hence unit digit be $x+2$.

therefore the number will be $10x+(x+2)=11x+2$----(1)

now $x^2+{(x+2)}^2-(11x-2)=\frac{2x(x+3)}{3}$

$2x^2+4x+4-11x-2=\frac{2x(x+2)}{3}$

$6x^2-21x+6=2x^2+4x$

$4x^2-25x+6=0$

$4x^2-24x-x+6=0$

$4x(x-6)-1(x-6)=0$

$(4x-1)(x-6)=0$

$x=\frac{1}{4}$ and $x=6$

now as $x$ has to be integer

hence $x=6$.

thus number is $11x+2=68$