A cube of white material is painted red on all its surfaces. If it is cut into 125 smaller cubes of same size, then how many cubes will have two sides painted red?

As cube is cut into 125 identical smaller cubes, we can say that size of larger cube is $$5\times5\times5$$. 

The cubes which will be having exactly 2 faces painted will be the cubes which lie on edges but not corners of the cube because corner cubes will have 3 faces painted. 

That is for every edge of the cube , out of "$$n$$" cubes there will be "$$n-2$$" cubes (excluding corners) which are painted on 2 faces exactly . So for the entire cube there are 12 edges , so total no of cubes with exactly 2 faces painted $$=12(n-2)$$. 

Therefore, in this case as$$n=5$$, the total no of cubes with exactly 2 faces painted white$$=12\left(5-2\right)=36$$.