Question

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

Answer 1

well definition of odd is 2k+1

Answer 2

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)

Answer 3

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 )
\[A =\frac{(m+3)^n+1}{3m} \in \mathbb{Z} \\ A(3m)=(m+3)^n+1 \\ 0=(m+3)^n+1-A(3m) \\ (m+3)^n+1-3Am=0\]

Answer 4

The given problem is equivalent to :
\[\dfrac{m^n+3^n+1}{3m} =A \in \mathbb{Z} \implies \text{A is odd}\]

Answer 5

you did mod 3

Answer 6

why does doing that given us an equivalent problem

Answer 7

\[(m+3)^n = m^n+3^n + 3m(\text{stuff})\]

Answer 8

Hmm yeah not exactly same as it disturbs the parity of \(A\)

Answer 9

actually found an answer but I don't like it because I don't understand it :p

Answer 10

do you want me to share it?

Answer 11

yeah im almost stuck haha

Answer 12

http://artofproblemsolving.com/community/c2202t169f2202_algebra

Answer 13

click "This LTE thing is pretty cool"

Answer 14

where I have looked up LTE and I believe it just means lifting the exponent lemma/theorem thingy

Answer 15

if m is even why does that allow us to do mod 3?

Answer 16

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 )

Answer 17

click \(\href{https:///www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=20&cad=rja&uact=8&ved=0CFQQFjAJOAo&url=http%3A%2F%2Fwww.math.muni.cz%2F~bulik%2Fvyuka%2Fpen-20070711.pdf&ei=suSIVZ-ZL4G9sAWMh7QY&usg=AFQjCNESVFKertjKo1RRDenoVvOsgsul0w}{this}\)

Answer 18

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 )

Answer 19

what about bit.ly sites?

Answer 20

omg too cute
and it was A17 just in case you were scrolling for the probem

Answer 21

@UsukiDoll what is that "bit.ly sites"?

Answer 22

or how do you use it

Answer 23

@freckles change http:// to http:///

3 forward slashes are needed

Answer 24

bit.ly <--- online site used to shorten long url links

Answer 25

\[\href{
http:///www.wolframalpha.com/input/?i=integrate%28kitty%2Ckitty%29
}{WOLFRAM!}
\]

Answer 26

congrats!!

Answer 27

you can teach an old dog new tricks
so congrats to you on doing just that :p

Answer 28

http://bit.ly/1SFjO6a
this is the thing @UsukiDoll was talking about

Answer 29

That is kind of neat too

Answer 30

though I kind of like writing my links in cute words

Answer 31

bit.ly <--- online site used to shorten long url links

Answer 32

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