Which of the following IPv$4$ addresses is a valid IPv$4$ address and has the same network ID as $192\mathrm{.}168\mathrm{.}112\mathrm{.}45$ given the subnet mask $255\mathrm{.}255\mathrm{.}255\mathrm{.}0?$

a$.172\mathrm{.}16\mathrm{.}112\mathrm{.}45$

b$.192\mathrm{.}168\mathrm{.}112\mathrm{.}257$

c$.10\mathrm{.}0\mathrm{.}0\mathrm{.}45$

d$.192\mathrm{.}168\mathrm{.}113\mathrm{.}46$

e$.192\mathrm{.}168\mathrm{.}112.$