# Two consecutive numbers are relatively prime

• Jun 20th 2010, 11:19 AM
dwsmith
Two consecutive numbers are relatively prime
Two consecutive numbers are relatively prime.

$\displaystyle (n, n+1)=1\rightarrow \alpha n+\beta (n+1)=1$

I am not sure what to do next to prove this.
• Jun 20th 2010, 11:20 AM
chiph588@
Quote:

Originally Posted by dwsmith
Two consecutive numbers are relatively prime.

$\displaystyle (n, n+1)=1\rightarrow \alpha n+\beta (n+1)=1$

I am not sure what to do next to prove this.

$\displaystyle \alpha=-1,\; \beta=1$
• Jun 20th 2010, 11:23 AM
dwsmith
I see how that will justify the equality but is that ok to do for the proof and then just call it day and it is done?
• Jun 20th 2010, 01:05 PM
chiph588@
Quote:

Originally Posted by dwsmith
I see how that will justify the equality but is that ok to do for the proof and then just call it day and it is done?

Let $\displaystyle g=(n,n+1)$

$\displaystyle g\mid n$ and $\displaystyle g\mid n+1 \implies g\mid (n+1)-n = 1 \implies g=\pm1$. But since $\displaystyle g>0$, we get $\displaystyle g=1$.
• Jun 20th 2010, 02:35 PM
melese
Quote:

Originally Posted by dwsmith
I see how that will justify the equality but is that ok to do for the proof and then just call it day and it is done?

The answer is yes. In general: Let $\displaystyle a$ and $\displaystyle b$ be integers. There exists two integers $\displaystyle x$ and $\displaystyle y$ such that $\displaystyle ax+by=1$ if and only if $\displaystyle (a,b)=1$. In your case, $\displaystyle a=n$, $\displaystyle x=-1$ and $\displaystyle b=n +1$, $\displaystyle y=1$.

However, the reply you got from "chiph588" using the greatest common divisor ($\displaystyle g=(n,n+1)$) seems to me more elegant.
• Jun 20th 2010, 02:50 PM
chiph588@
Quote:

Originally Posted by melese
The answer is yes. In general: Let $\displaystyle a$ and $\displaystyle b$ be integers. There exists two integers $\displaystyle x$ and $\displaystyle y$ such that $\displaystyle ax+by=1$ if and only if $\displaystyle (a,b)=1$. In your case, $\displaystyle a=n$, $\displaystyle x=-1$ and $\displaystyle b=n +1$, $\displaystyle y=1$.

This is a special case of Bézout's identity.