Show that $\displaystyle (m,n) = 1 \Rightarrow (m) \cap (n) = (mn) $. so $\displaystyle mx+ny = 1 $. Now $\displaystyle (m) = \{cm: c \in R \} $ and $\displaystyle (n) = \{dm: d \in R \} $. So show that there is some common element which implies that they must be equal?