Problem 4:
$(m,n)$ represents the largest common divisor of integers $m$ and $n$. For example $(2,3)=1$ and $(10,15)=5$. Suppose $n(n+1)(n+2)$ is a square, where $n$=integer.
a) What is $(n,n+1)$?
b) What is $(n+1,n+2)$?
c) What is $(n+1,n(n+2))$?
From your answers $a,b,c$ is it possible for $n(n+1)(n+2)$ to be a square?
BdMO National Higher Secondary 2007/4
- Cryptic.shohag
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Re: BdMO National Higher Secondary 2007/4
n, (n+1), (n+2) are back to back integers. If n is even, then (n+1) is odd and (n+2) is even. And if n is odd, then (n+1) is even and (n+2) is odd. So, (n+1) is coprime to both n and (n+2). So....
a) (n,n+1)=1
b) (n+1,n+2)=1
c) (n+1, n(n+2))=1
Now, as there is no comon factor between (n+1) and n(n+2), n(n+1)(n+2) cannot be a square....
a) (n,n+1)=1
b) (n+1,n+2)=1
c) (n+1, n(n+2))=1
Now, as there is no comon factor between (n+1) and n(n+2), n(n+1)(n+2) cannot be a square....
God does not care about our mathematical difficulties; He integrates empirically. ~Albert Einstein
Re: BdMO National Higher Secondary 2007/4
There is no common factor between $2^2,3^2$ but their product is $6^2$
One one thing is neutral in the universe, that is $0$.
- Cryptic.shohag
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Re: BdMO National Higher Secondary 2007/4
$2^2,3^2$ are not back to back integers but n,(n+1) are. Putting square makes difference here....Masum wrote:There is no common factor between $2^2,3^2$ but their product is $6^2$
God does not care about our mathematical difficulties; He integrates empirically. ~Albert Einstein
Re: BdMO National Higher Secondary 2007/4
Actually Masum wanted to say about this logic. When you use some logic it should be true in all cases. I mean you said that $gcd(n,n+1)=1 \Longrightarrow n(n+1)(n+2)$ is not a square, your logic sound something like $gcd(a,b)=1 \Longrightarrow ab(\text{something relatively prime to b})$ can not be a square.Cryptic.shohag wrote:Now, as there is no comon factor between (n+1) and n(n+2), n(n+1)(n+2) cannot be a square....
And your solution has a few problems (that are fixable easily, though). The line I quoted is not correct in all cases.
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
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Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
- Cryptic.shohag
- Posts:16
- Joined:Fri Dec 17, 2010 11:32 pm
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Re: BdMO National Higher Secondary 2007/4
Actually my logic was if a and b r back to back common integers then $(a,b)=1$ and ab can't be a square. Is there any problem in my logic moon vaia? If so, plz give me an example....
God does not care about our mathematical difficulties; He integrates empirically. ~Albert Einstein