Question

If difference between two positive integers is $2$ and the difference between their cubes is $56$, then the numbers are $2$ and $4$.

State whether the above statement is true or false.

• A. True
• B. False

Answer 1

The correct option is A True

Let the two numbers be $a$ and $b$
$a-b=2$ or $a=b+2$
$a^3-b^3=56$
$(b+2{)}^3-b^3=56$
$b^3+8+6b^2+12b-b^3=56$
$6b^2+12b-48=0$
$b^2+2b-8=0$
$b^2+4b-2b-8=0$
$\Rightarrow b=-4,2$
Hence, $a=b+2=4$
Thus the numbers are $2$ and $4$.