8^n - 3^n is divisible by 5

Question

My compitations: Base Case= $8-3 = 5$ is divisible by $5$ Inductive Hypothesis$=$ $8^k - 3^k = 5M$ $8^k = 5M + 3^k$ $3^k = 8^k - 5M$

My final answer is $5(11M + 11^k)$ Please tell me if I have any mistake regarding my computations.

Answer 1

note that $$8\equiv 3\mod 5$$ thus $$8^n\equiv 3^n\mod 5$$ and therefore $$8^n-3^n\equiv 3^n-3^n=0\mod 5$$

Answer 2

Hint: $8^{n+1}-3^{n+1}=8\cdot8^{n}-3\cdot3^{n}=(5+3)\cdot8^{n}-3\cdot3^{n}=5\cdot8^{n}+3\cdot(8^{n}-3^{n})$

More generally, if $a,b \in \mathbb Z$ then $d=a-b$ divides $a^{n}-b^{n}$ for all $n$ because $a^{n+1}-b^{n+1}=d\cdot a^{n}+b\cdot(a^{n}-b^{n})$.

Answer 3

You can also use:

$$ \forall n \geq 1, \forall x, y \in \mathbb{R}, x^n -y^n = (x-y)\sum_{k=0}^{n-1} x^k y^{n-1-k}$$

Hence :

$$ 8^n -3^n = (8-3)\sum_{k=0}^{n-1} 8^k 3^{n-1-k}$$

$$ =5\times K$$

where $K \in \mathbb{N}$.