The figures above show a hexagonal nut that has a width of 1~$\frac{5}{16}$~ inches and a wrench that, in order to fit the nut. must have a width of at least 1~$\frac{5}{16}$~ inches. Of all the wrenches that fit the nut and have widths that are whole numbers of millimeters, the wrench that fits the nut most closely has a width of how many millimeters?

(Note: 1 inch ≈ 25. 4 millimeters)