freckles
  • freckles
\[\text{ Let } m \text{ and } n \text{ be natural numbers such that } \\ A=\frac{(m+3)^n+1}{3m} \text{ is an integer } \\ \text{ Show } A \text{ is odd.} \]
Mathematics
schrodinger
  • schrodinger
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UsukiDoll
  • UsukiDoll
well definition of odd is 2k+1
freckles
  • freckles
Assume A is even then A=2i where i is an integer. \[2i=\frac{(m+3)^n+1}{3m} \\ \text{ let's see if we can arrive at some contradiction } \\ 2i(3m)=(m+3)^n+1 \\ 2(3im)=(m+3)^n+1 \text{ so } (m+3)^n+1 \text{ is even } \\ 2(3im)-1 \text{ is odd } \text{ so } (m+3)^n \text{ is odd } \] thinking a bit more... (not sure if I'm going to get this to work)
freckles
  • freckles
having trouble with contradiction ...was thinking about writing it like this but I don't see how to use vieta's jumping of whatever it is called if I can even use that ) \[A =\frac{(m+3)^n+1}{3m} \in \mathbb{Z} \\ A(3m)=(m+3)^n+1 \\ 0=(m+3)^n+1-A(3m) \\ (m+3)^n+1-3Am=0\]

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ganeshie8
  • ganeshie8
The given problem is equivalent to : \[\dfrac{m^n+3^n+1}{3m} =A \in \mathbb{Z} \implies \text{A is odd}\]
freckles
  • freckles
you did mod 3
freckles
  • freckles
why does doing that given us an equivalent problem
ganeshie8
  • ganeshie8
\[(m+3)^n = m^n+3^n + 3m(\text{stuff})\]
ganeshie8
  • ganeshie8
Hmm yeah not exactly same as it disturbs the parity of \(A\)
freckles
  • freckles
actually found an answer but I don't like it because I don't understand it :p
freckles
  • freckles
do you want me to share it?
ganeshie8
  • ganeshie8
yeah im almost stuck haha
freckles
  • freckles
http://artofproblemsolving.com/community/c2202t169f2202_algebra
freckles
  • freckles
click "This LTE thing is pretty cool"
freckles
  • freckles
where I have looked up LTE and I believe it just means lifting the exponent lemma/theorem thingy
freckles
  • freckles
if m is even why does that allow us to do mod 3?
freckles
  • freckles
By the way I got my problem from here if you wanted to know : https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=20&cad=rja&uact=8&ved=0CFQQFjAJOAo&url=http%3A%2F%2Fwww.math.muni.cz%2F~bulik%2Fvyuka%2Fpen-20070711.pdf&ei=suSIVZ-ZL4G9sAWMh7QY&usg=AFQjCNESVFKertjKo1RRDenoVvOsgsul0w (by the way do you know if there is a way to hide how long this link actually is using some kind of LaTeX voodoo )
ganeshie8
  • ganeshie8
click \(\href{https:///www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=20&cad=rja&uact=8&ved=0CFQQFjAJOAo&url=http%3A%2F%2Fwww.math.muni.cz%2F~bulik%2Fvyuka%2Fpen-20070711.pdf&ei=suSIVZ-ZL4G9sAWMh7QY&usg=AFQjCNESVFKertjKo1RRDenoVvOsgsul0w}{this}\)
freckles
  • freckles
By the way I got my problem from here if you wanted to know : https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=20&cad=rja&uact=8&ved=0CFQQFjAJOAo&url=http%3A%2F%2Fwww.math.muni.cz%2F~bulik%2Fvyuka%2Fpen-20070711.pdf&ei=suSIVZ-ZL4G9sAWMh7QY&usg=AFQjCNESVFKertjKo1RRDenoVvOsgsul0w (by the way do you know if there is a way to hide how long this link actually is using some kind of LaTeX voodoo )
UsukiDoll
  • UsukiDoll
what about bit.ly sites?
freckles
  • freckles
omg too cute and it was A17 just in case you were scrolling for the probem
freckles
  • freckles
@UsukiDoll what is that "bit.ly sites"?
freckles
  • freckles
or how do you use it
ganeshie8
  • ganeshie8
@freckles change http:// to http:/// 3 forward slashes are needed
UsukiDoll
  • UsukiDoll
bit.ly <--- online site used to shorten long url links
freckles
  • freckles
\[\href{ http:///www.wolframalpha.com/input/?i=integrate%28kitty%2Ckitty%29 }{WOLFRAM!} \]
ganeshie8
  • ganeshie8
congrats!!
freckles
  • freckles
you can teach an old dog new tricks so congrats to you on doing just that :p
freckles
  • freckles
http://bit.ly/1SFjO6a this is the thing @UsukiDoll was talking about
freckles
  • freckles
That is kind of neat too
freckles
  • freckles
though I kind of like writing my links in cute words
UsukiDoll
  • UsukiDoll
bit.ly <--- online site used to shorten long url links
freckles
  • freckles
hmmm anyways a mod 3 =0,1, or 2 1 mod 3=1 2 mod 3=2 3 mod 3=0 4 mod 3=1 5 mod 3=2 6 mod 3=0 ... 1,4,...,1+3i mod 3 will give us 1 2,5,...,2+3i mod 3 will give us 2 3,6,...,3+3i mod 3 will give us 3 (which this makes sense) but any of 1+3i,2+3i,3+3i can be even or odd so I don't understand if we assume m is even why we can do mod 3 like what does it tell us doing that

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