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**tttcomrader** Let $\displaystyle G = \frac { \mathbb {Q} } { \mathbb {Z} } $ under +, so the elements are the equivalence classes $\displaystyle \hat {r} = \{ s \in \mathbb {Q} : s-r \in \mathbb {Z} \} $. Write $\displaystyle r \equiv s \ (mod \ 1 ) $ if $\displaystyle r - s \in \mathbb {Z} $.

Prove that if $\displaystyle r = \frac {a}{b} $ with $\displaystyle a,b \in \mathbb {Z} $, $\displaystyle gcd (a,b) = 1 $ and $\displaystyle b > 0 $, then $\displaystyle \hat {r} $ has order b and $\displaystyle < \hat {r} > = < \hat { \frac {1}{b} } > $