are two consecutive numbers relatively prime?
I have a question.
I have been given this proof: "For any $n$ in the integers where $n>2$, show there are at least $2$ elements in $U(n)$ that satisfy $x^2=1$."
I have gone through and actually proved this, (that the numbers are $1$ and $n-1$) but i didn't' know how to prove that $n-1$ is in fact in the set $U(n)$. Is it because two consecutive numbers are always relatively prime?
Solution 1:
$n$ is coprime to $n-1$, for if $d$ divides $n$ and $d$ divides $n-1$, then $d$ divides $n-(n-1)=1$.