Geometry + word problem

Aman verma wrote:Q: A cuboid of dimensions 51, 85 , 102 cm is first painted by red colour then it is cut into minimum possible identical cubes.Now the total surface area of all those faces of cubes which are not red is :

a) 119646 cm^2

b) 52020 cm^2

c) 18514 cm^2

d) 36414 cm^2

e) 58416 cm^2

Calculate the total volume. 442170
Then dividing by identical cubes

What has volume got to do with surface area ?

Aman verma wrote:Q: A cuboid of dimensions 51, 85 , 102 cm is first painted by red colour then it is cut into minimum possible identical cubes.Now the total surface area of all those faces of cubes which are not red is :

a) 119646 cm^2

b) 52020 cm^2

c) 18514 cm^2

d) 36414 cm^2

e) 58416 cm^2

There is no minimum possible there unless there is an integer constraint - Do the sides have to be integers for the cubes formed??

Solution:
Since 51 = 17 X 3
85 = 17X 5
102= 17X 6

Therefore, the minimum possible number of cubes = 3X5X6 = 90
The total surface area of non red faces= Total surfaces of all the cubes - Total surface area of cuboid
=> (90 X 6X17X17 ) - 2X (51X85+85X102+102X51) [Surface area of a cube = 6a^2]
= 119646 cm^2
The answer is A.