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How many positive integers $$n$$ satisfy the condition $$2^n + n \mid 8^n + n$$?

Note by Sanjib Kumar Tripathy
6 days, 17 hours ago

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@Calvin Lin- could you please give an entire solution to this? · 11 hours ago

i did it already. please post the entire solution.i am quite...disturbed by this probelem · 12 hours ago

@Calvin Lin-i really have no idea how to go further. Could you please post a solution to this? · 12 hours ago

$$2^n + n \mid 8^n + n \Leftrightarrow 2^n + n \mid (2^n + n ) \times 4^n - n \times 4^n + n \Leftrightarrow 2^n + n \mid n \times 4^n - n$$ was what I thought you said you were doing earlier.

So we were able to remove the $$8^n$$ and replaced it with a $$n \times 4 ^n$$. How can we remove the $$4^n$$ now to further simplify the statement? Staff · 12 hours ago

@Calvin Lin-so sorry! it is 2^n+n| 2^n(2^n+1)(2^n-1) · 12 hours ago

That's a good start. Actually, $$8^n$$ is "more complicated" than $$n$$. How can we "eliminate exponents" to only obtain polynomials? Staff · 12 hours ago

@Calvin Lin-sir, please wait because i need to get the answer today. I dnt have anyone here else to help me out. · 12 hours ago

@Calvin Lin-2^n+n|(2^n-1)(2^n +n ) · 12 hours ago

Can you recheck? You are saying $$A \mid BA$$ where $$A = 2^n + n$$.

This doesn't quite look right to me. I now do not know what you are doing. Please list out your steps carefully, and do not just jump to the final step. Staff · 12 hours ago

@Calvin Lin-i did it already, multiplied and subtracted to to get (2^n-1)(2^n +n )|2^n+n. all i dnt know is how to get a polynomial out of it · 13 hours ago

Are you sure the RHS is 2^n+n? That seems to be the LHS. At this point in time, I'm not certain what you're doing. Staff · 13 hours ago

i did it already, multiplied and subtracted to to get (2^n-1)(2^n+n )|2^n+n. all i dnt know is how to get a polynomial out of it · 1 day, 5 hours ago

@Calvin Lin-can you make me understand how to get a polynomial from this? · 2 days, 1 hour ago

You already have the ideas, but just are not executing on them. You saw how to "simplify" $$8^n$$ into other terms, and just need to "simplify" these other terms into polynomials.

IE We have a $$2^{2n} ( \times n)$$ term. How can we get rid of it? Staff · 1 day, 13 hours ago

(2^n-1)(2^n-1)|2^n+n · 3 days, 1 hour ago

(2^2n)n-n|2^n+n · 3 days, 12 hours ago

Great. So once again, " $$2^{2n} \times n$$ is more complicated than $$2^n$$". How can we simplify further? Staff · 3 days, 12 hours ago

i tried that too. I multiplied 2^2n to 2^n+n and subtracted 8^n+n from that and proceeded as above. What i dont get is what to do next. · 3 days, 22 hours ago

What is the equation that you get? How can we simplify it further? Staff · 3 days, 13 hours ago

@CalvinLin-this was my approach. If 2^n+n devides 8^n+n then it devides 8^n+n-2^n+n.I wanted to eliminate n. Now i arrive at finding 2^n+n devides (2^n)(2^n+1)(2n-1). Here i am stuck to find the answer · 4 days, 17 hours ago

That's a good start. Actually, $$8^n$$ is "more complicated" than $$n$$. How can we "eliminate exponents" to only obtain polynomials? Staff · 4 days, 14 hours ago

i tried dat way too. I believe i am leaving a minute detail. Can ya please send me a solution? · 5 days, 20 hours ago

You can write up your solution and obtain feedback from the community on how to improve it. Staff · 5 days, 14 hours ago

through brute force 1,2 come. but what canbe a logical way to do this? · 6 days, 7 hours ago

I tried. cant get a clue to start · 6 days, 7 hours ago

Since we have a Diophantine equation, which approach can we use? Staff · 6 days, 7 hours ago