What is the least possible number of piece which have at most one colour on them?

For least number of cuts 120 = 4 x 5 x 6 i.e number of cuts must be 3, 4 and 5 in three planes in this case number of cubes on a face is either 6 x 5 = 30 or 6 x 4 = 24 or 4 x 5 = 20 cubes . And number of cuboids on an edge is 4 or 5 or 6
To satisfy this case all the cuboids on the edges and corners must have more than one colour on them. And in that case opposite face must have painted in the same colour.
In that case number of cuboids with 3 colours on them = 8
In that case number of cuboids with 2 colours on them = 4 x (2 + 3 + 4 ) = 36
Hence number of cuboids with at least 1 colour on them is 120 - 36 - 8 = 76