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freckles
 one year ago
\[\text{ Let } m \text{ and } n \text{ be natural numbers such that } \\ A=\frac{(m+3)^n+1}{3m} \text{ is an integer } \\ \text{ Show } A \text{ is odd.} \]
freckles
 one year ago
\[\text{ Let } m \text{ and } n \text{ be natural numbers such that } \\ A=\frac{(m+3)^n+1}{3m} \text{ is an integer } \\ \text{ Show } A \text{ is odd.} \]

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UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0well definition of odd is 2k+1

freckles
 one year ago
Best ResponseYou've already chosen the best response.1Assume 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
 one year ago
Best ResponseYou've already chosen the best response.1having 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 ) \[A =\frac{(m+3)^n+1}{3m} \in \mathbb{Z} \\ A(3m)=(m+3)^n+1 \\ 0=(m+3)^n+1A(3m) \\ (m+3)^n+13Am=0\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1The given problem is equivalent to : \[\dfrac{m^n+3^n+1}{3m} =A \in \mathbb{Z} \implies \text{A is odd}\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.1why does doing that given us an equivalent problem

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1\[(m+3)^n = m^n+3^n + 3m(\text{stuff})\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1Hmm yeah not exactly same as it disturbs the parity of \(A\)

freckles
 one year ago
Best ResponseYou've already chosen the best response.1actually found an answer but I don't like it because I don't understand it :p

freckles
 one year ago
Best ResponseYou've already chosen the best response.1do you want me to share it?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1yeah im almost stuck haha

freckles
 one year ago
Best ResponseYou've already chosen the best response.1http://artofproblemsolving.com/community/c2202t169f2202_algebra

freckles
 one year ago
Best ResponseYou've already chosen the best response.1click "This LTE thing is pretty cool"

freckles
 one year ago
Best ResponseYou've already chosen the best response.1where I have looked up LTE and I believe it just means lifting the exponent lemma/theorem thingy

freckles
 one year ago
Best ResponseYou've already chosen the best response.1if m is even why does that allow us to do mod 3?

freckles
 one year ago
Best ResponseYou've already chosen the best response.1By 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%2Fpen20070711.pdf&ei=suSIVZZL4G9sAWMh7QY&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
 one year ago
Best ResponseYou've already chosen the best response.1click \(\href{https:///www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=20&cad=rja&uact=8&ved=0CFQQFjAJOAo&url=http%3A%2F%2Fwww.math.muni.cz%2F~bulik%2Fvyuka%2Fpen20070711.pdf&ei=suSIVZZL4G9sAWMh7QY&usg=AFQjCNESVFKertjKo1RRDenoVvOsgsul0w}{this}\)

freckles
 one year ago
Best ResponseYou've already chosen the best response.1By 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%2Fpen20070711.pdf&ei=suSIVZZL4G9sAWMh7QY&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
 one year ago
Best ResponseYou've already chosen the best response.0what about bit.ly sites?

freckles
 one year ago
Best ResponseYou've already chosen the best response.1omg too cute and it was A17 just in case you were scrolling for the probem

freckles
 one year ago
Best ResponseYou've already chosen the best response.1@UsukiDoll what is that "bit.ly sites"?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1@freckles change http:// to http:/// 3 forward slashes are needed

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0bit.ly < online site used to shorten long url links

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[\href{ http:///www.wolframalpha.com/input/?i=integrate%28kitty%2Ckitty%29 }{WOLFRAM!} \]

freckles
 one year ago
Best ResponseYou've already chosen the best response.1you can teach an old dog new tricks so congrats to you on doing just that :p

freckles
 one year ago
Best ResponseYou've already chosen the best response.1http://bit.ly/1SFjO6a this is the thing @UsukiDoll was talking about

freckles
 one year ago
Best ResponseYou've already chosen the best response.1That is kind of neat too

freckles
 one year ago
Best ResponseYou've already chosen the best response.1though I kind of like writing my links in cute words

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0bit.ly < online site used to shorten long url links

freckles
 one year ago
Best ResponseYou've already chosen the best response.1hmmm 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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