Geometry task from high school competition

[Nekemtenee]

Okay, so I was in a math competition. I could deal with a lot of tasks, but I had a task I couldn't do ( I hadn't enough time... I got 10 minutes for one) So let's see the task:
we made a 4cmx6cmx10cm cuboid from 240 pieces of 1 cm side-edged cubes. How many cuboids we can do from these small cubes? (the cuboid's side-edges must be parallels with the original cuboid's side)

Any idea to solve it correctly? In my opinion a small cube is also a cuboid so at least there is 240... And I have other ideas too, but I can't deal with it completely.

 

[Ishuda]

Okay, so I was in a math competition. I could deal with a lot of tasks, but I had a task I couldn't do ( I hadn't enough time... I got 10 minutes for one) So let's see the task:
we made a 4cmx6cmx10cm cuboid from 240 pieces of 1 cm side-edged cubes. How many cuboids we can do from these small cubes? (the cuboid's side-edges must be parallels with the original cuboid's side)

Any idea to solve it correctly? In my opinion a small cube is also a cuboid so at least there is 240... And I have other ideas too, but I can't deal with it completely.

The maximum volume is 240 cm³ and, since the building blocks are 1 cm³ it seems like the number would be the permutations of the factors of 240:
240 = 2⁴ * 3 * 5
so we have
240 X 1 X 1
120 X 2 X 1
60 X 4 X 1
60 X 2 X 2
...
or just use the proper formula.

Edit to add: You can also read those as 240 (1 X 1)'s, 120 (2 X 1)'s, etc.

 

[soroban]

Hello Nekemtenee!

We made a 4x6x10 cuboid from 240 pieces of 1 cm side-edged cubes.
How many cuboids we can do from these small cubes?

Factor 240 into 3 factors.. . \(\displaystyle \begin{array}{c}1\cdot1\cdot240 \\ 1\cdot2\cdot120 \\ 1\cdot3\cdot80 \\ 1\cdot4\cdot60 \\ 1\cdot5\cdot48\\ 1\cdot6\cdot40 \\ 1\cdot8\cdot30 \\ 1\cdot10\cdot24 \\ 1\cdot12\cdot20 \\ 1\cdot15\cdot16 \end{array} \quad \begin{array}{c} 2\cdot2\cdot60 \\ 2\cdot3\cdot40 \\ 2\cdot4\cdot30\\ 2\cdot5\cdot24 \\ 2\cdot6\cdot20 \\ 2\cdot8\cdot15 \\ 2\cdot10\cdot2 \end{array} \qquad \begin{array}{c}3\cdot4\cdot20 \\ 3\cdot5\cdot16 \\ 3\cdot8\cdot10 \\ 4\cdot4\cdot15 \\ 4\cdot5\cdot12 \\ 4\cdot6\cdot10\\ 5\cdot6\cdot8 \end{array}\)

I find 24 possible cuboids.