In a Young's interference experiment, the two slits are separated by $0\mathrm{.}140$ mm and the incident light includes two wavelengths: $1=540$ nm $($green$)$ and $2=450$ nm $($blue$).$ The overlapping interference patterns are observed on a screen $1\mathrm{.}56$ m from the slits.

$($a$)$ Find a relationship between the orders m$1$ and m$2$ that determines where a bright fringe of the green light coincides with a bright fringe of the blue light. $($The order m$1$ is associated with $1,$ and m$2$ is associated with $2.)$

m$2$m$1$

$=$

$($b$)$ Find the minimum values of m$1$ and m$2$ such that the overlapping of the bright fringes will occur.

m$1=$m$2=$

Find the position of the overlap on the screen.

cm from the central maximum