Question

Show that ${13}^3-5^3$ is divisible by 8.

Answer 1

Let $f\left(x\right)=x^n-y^n$
By remainder theorem,, if $f\left(a\right)=0$, then $x-a$ divides $f\left(x\right)$
Thus, $f\left(y\right)=y^n-y^n$
$=0$
Thus, $(x-y)$ divides $x^n-y^n$
Here, given polynomial is ${13}^3-5^3$
Comparing it with the above, $x=13$, $y=5$, and $n=3$
then, $(x-y)=13-5=8$
Using the above proof, we can say that ${13}^3-5^3$ is divisible by $8$