Assume that the shape of the objects makes it easy to divide the object into any smaller size, like a rectangular solid or cylinder. The shapes of the cut objects is not relevant, just their sizes.
As an example, suppose we want to divide 5 objects into 6 groups. Start by taking 3 of the objects and dividing each of them in half. Place each of the six halves in a different group. Next, take the 2 remaining objects and divide each one in thirds. Place each of the thirds into one of the 6 groups. The object now lies completely in the 6 groups and each group is identical, with a 1/2 object and a 1/3 object.
Was I able to do this because 5 = 3 + 2, and 3 and 2 divide evenly into 6? If so, is there a simple way of proving that this condition must be satisfied? What area of mathematics does this fall under?