Maths Question for smart

May 2016
in my house
Mr Greengold had received a consignment of gold in the form of one-centimetre cubes. To conceal them, he ordered that the cubes be glued together to form a solid cuboid and had all the outside faces of this cuboid painted camouflage green.

Jimmy Bold carefully spied on these proceedings and sent the following message to Headquarters:

Greengold has 1 000 000 000 one-centimetre cubes of gold glued together into a cuboid, the length, width and height of which, measured in centimetres, contain no zeros.'

a. Find one set of dimensions (Length, width and height) that satisfies this condition.

b. Now endeavour to find all possible sets of dimensions that satisfies the above condition.

He received a message back to say that this was not sufficient information to determine the dimensions of the cuboid. More spying revealed that the quantity of paint used was less than 8 tins, the paint in each tin covering 100 square metres.

With this additional information, Headquarters was able to tell the satellite surveillance people the precise dimensions of the cuboid so that they could locate it.

c. Explain why there is only one possible set of dimensions and find these dimensions.