An ionic solid MX crystallizes in a face-centered cubic lattice...

In a face-centered cubic lattice where anions form the lattice and cations occupy all octahedral voids, the structure is of the rock-salt ($\text{NaCl}$) type.

In this structure, the cations and anions touch each other along the edge of the unit cell.

The relationship between the edge length ($a$) and the ionic radii ($r_+$ and $r_-$) is given by:
$a = 2(r_+ + r_-)$

Given:
$a = 514\text{ pm}$
$r_- = 181\text{ pm}$

Substituting the given values:
$514 = 2(r_+ + 181)$
$257 = r_+ + 181$
$r_+ = 257 - 181 = 76\text{ pm}$

Answer: $76$