How to show $8^n-3^n$ is divisible by $5$ using the principle of induction? [closed]

I am stuck on one question:

Show that $8^n-3^n$ is divisible by $5$.

Thank you

-

closed as off-topic by Jonas Meyer, N. F. Taussig, Alan, Adam Hughes, 2mkgzMar 24 at 4:01

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jonas Meyer, N. F. Taussig, Alan, Adam Hughes, 2mkgz
If this question can be reworded to fit the rules in the help center, please edit the question.

What have you tried? Try factoring. –  Graphth Mar 13 '12 at 2:41
I'm agree with Graphth –  Lrrr Apr 8 '12 at 15:58
In order to receive help, it is always best form to show what you tried and not expect somebody to just give you an answer. –  agent154 Sep 20 '13 at 3:31

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.

-

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

-

In case you want to improve your basic knowledge about mathematics, it is good to know about Modular arithmetic. with a little knowledge in that field you will know that $8\equiv 3 \mod{5}$.

A simpler(to understand) solution if you didn't understand module :

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 in them so : $$8^n-3^n = (5*k+3^n)-3^n = 5k$$

-
@RebeccaJ.Stones This answer was posted three years ago. –  columbus8myhw Mar 24 at 2:45
Oh, I didn't realize. The question came up in the review queue, so I reviewed it. –  Rebecca J. Stones Mar 24 at 2:47
@RebeccaJ.Stones Thanks for your comment, I'll change the link :) –  Lrrr Mar 24 at 20:54