A help here please
How many positive integers satisfy the condition ?
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4 years, 5 months ago
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What have you tried? Where are you stuck?
I tried. cant get a clue to start
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Since we have a Diophantine equation, which approach can we use?
through brute force 1,2 come. but what canbe a logical way to do this?
i tried dat way too. I believe i am leaving a minute detail. Can ya please send me a solution?
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@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
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That's a good start. Actually, 8n is "more complicated" than n. How can we "eliminate exponents" to only obtain polynomials?
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.
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What is the equation that you get? How can we simplify it further?
(2^2n)n-n|2^n+n
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Great. So once again, " 22n×n is more complicated than 2n". How can we simplify further?
(2^n-1)(2^n-1)|2^n+n
@Calvin Lin-can you make me understand how to get a polynomial from this?
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You already have the ideas, but just are not executing on them. You saw how to "simplify" 8n into other terms, and just need to "simplify" these other terms into polynomials.
IE We have a 22n(×n) term. How can we get rid of it?
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
@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
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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.
@Calvin Lin-2^n+n|(2^n-1)(2^n +n )
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Can you recheck? You are saying A∣BA where A=2n+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.
@Calvin Lin-sir, please wait because i need to get the answer today. I dnt have anyone here else to help me out.
@Calvin Lin-so sorry! it is 2^n+n| 2^n(2^n+1)(2^n-1)
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That's a good start. Actually, 8n is "more complicated" than n. How can we "eliminate exponents" to only obtain polynomials?
@Calvin Lin-i really have no idea how to go further. Could you please post a solution to this?
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2n+n∣8n+n⇔2n+n∣(2n+n)×4n−n×4n+n⇔2n+n∣n×4n−n was what I thought you said you were doing earlier.
So we were able to remove the 8n and replaced it with a n×4n. How can we remove the 4n now to further simplify the statement?
i did it already. please post the entire solution.i am quite...disturbed by this probelem
@Calvin Lin- could you please give an entire solution to this?