Question

$a^n-b^n$ is always divisible by

• A. 2a-b
• B. a+b
• C. a-b
• D. a-2b

Answer 1

The correct option is C

a-b

Substituting n=1, we get
$a^1-b^1=a-b$
Let $P\left(n\right):a^n-b^n\text{is divisible by a-b}$
P(1) is true
Assume P(k) is true
$a^k-b^k\text{is divisible by a-b}{a}^k-b^k=m(a-b)Now,a^{k+1}-b^{k+1}=a.(a^k-b^k)+(a-b)b^k=(am+b^k)(a-b)\to \text{divisible by a-b}$
P(k+1) is true

Hence, P(n) is true.