Induction principle for verifying divisibility

Question

I am stuck on one question: Show that $8^n-3^n$ is divisible by $5$.

Thank you

Answer 1

If n=1, we have $8^1-3^1=5$, which is divided by $5$.

Now, assume that $8^n - 3^n$ is divisible by $5$, i.e. $8^n - 3^n = 5k$ for some $k\in N$. Then $8^n = 5k + 3^n$. Thus, for $n + 1$, we have $8^{n+1} - 3^{n+1} = 8 · 8^n - 3 · 3^n = 8(5k + 3^n) - 3 · 3^n = 8 · 5k + 5 · 3^n = 5(8k + 3^n)$. Since $8k + 3^n$ is integer, we obtain that $8^{n+1} - 3^{n+1}$ is divisible by $5$. So, we proved the induction step. Thus we proved the statement.

Answer 2

Hint $\rm\ \ \ 8^{N+1}-\:3^{N+1} =\: (8-3)\:8^N + 3\:(8^N-3^N).\ $ This is prototypical telescopy.

Answer 3

In the case you want to improve your basic knowledge about mathematics, it is good to know about module. if you read this link you will know that $8\equiv 3 \mod{5}$.

or even simpler solution if you didn't understand module is this :

consider 8 = 5+3 so: $$8^n = (5+3)^n = 5^n + \dbinom{n}{1} * 5^{n-1} * 3 + ... + \dbinom{n}{n-1} * 5 * 3^{n-1} + 3^n = 5*k+3^n$$(as you can see n first term have 5)$$8^n-3^n = (5*k+3^n)-3^n = 5k$$