2b) Prove or disprove the following statement. (2) “If m divides a n −b n , then m divides abn −ban also.”

Solution:

Given that $m$ divides $a^n-b^n$ .

We know that for any integer $n$,

$a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})$ ...(i)

$\Rightarrow m$ divides $(a-b)$.

Now, consider $ab^n-ba^n$

$=-ab(a^{n-1}-b^{n-1})$

$=-ab(a-b)(a^{n-2}+a^{n-3}b+...+ab^{n-3}+b^{n-2})$ [Using (i)]

We can see that $(a-b)$ is also a factor here and thus, divisible by $m$.

Hence, yes, $m$ divides $ab^n-ba^n$.

Proved.