Question

Show that $m-1$ is a factor of $m^{21}-1$ and $m^{22}-1$.

Answer 1

Let $f\left(m\right)=m^{21}-1$ and $g\left(m\right)=m^{22}-1$.

The remainders when $f\left(m\right)$ and $g\left(m\right)$ are divided by $m-1$ are $f\left(1\right)$ and $g\left(1\right)$ respectively.

Now $f\left(1\right)=0=g\left(1\right)$.

Now by factor theorem, as the remainders are zero, so the $f\left(m\right)$ and $g\left(m\right)$ are perfectly divisible by $m-1$.

So $m-1$ is a factor of the given expressions.