Question

The difference between two positive integers is $2$ and the difference between their cubes is $56$. Find the numbers.

Answer 1

Let the numbers be x and y.

According to question

$x-y=2$ …………..$\left(1\right)$

$x^3-y^3=56$ …………..$\left(2\right)$

From $\left(1\right)$

$x=y+2$

Substitute the value of x in equation $\left(2\right)$

$(y+2{)}^3-y^3=56$

$\Rightarrow y^3+8+6y^2+12y-y^3=56$ .................. Using $(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$

$\Rightarrow 6y^2+12y-56+8=0$

$\Rightarrow 6y^2+12y-48=0$

$\Rightarrow 6(y^2+2y-8)=0$

$\Rightarrow y^2+4y-2y-8=0$

$\Rightarrow y(y+4)-2(y+4)=0$

$\Rightarrow (y+4)(y-2)=0$

$y+4=0$ $y-2=0$

$y=-4$ $y=2$

Y cannot be negative

$\therefore y=2$

Then $x=y+2$

$\Rightarrow x=2+2$

$\Rightarrow x=4$.