Consider a group of relatively prime positive integers that are less than the integer $n$. The operation is multiplication mod $n$

Show that there exist at least two integers in the group such that $x^2=1$.

assume that $n\gt 2$

Question

Consider a group of relatively prime positive integers that are less than the integer $n$. The operation is multiplication mod $n$

Show that there exist at least two integers in the group such that $x^2=1$.

assume that $n\gt 2$

ganeshie8 · Answer

for $n=5$, we get the set of relatively prime positive integers $\{1,2,3,4\}$
$1^2=1$ and $4^2=1$ are the two elements that satisfy the given condition

anonymous · Answer

Latex tip :) $$x^2\stackrel{n}{\equiv} 1$$

anonymous · Answer

In group {1,2,.......,n-1} , 1 is identity and 1^2=1 is always in the group.
We know that (n-1) is relative prime with n, that is gcd (n-1,n)=1 or $n-1\equiv 1(mod n)$
$(n-1)(n-1) \equiv 1*1=1(mod n)$
we can use multiplication table of group to show that there is at least 2 elements on the table satisfy the condition. 
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