This might be a stupid question but:

Assume you start with 1 stack of *n* items and want to end with *n* stacks with 1 item in each. To do this you successively divide it into smaller stacks with *a* and *b* items in each and create a product *ab*. (*n* is a natural number) Prove that the sum of all *ab* is equal to *n*(*n*\-1)/2.

Can I assume that the sum of all *ab* is the same no mater how you divide the stacks.

example *n* = 4

1. (4)

1. both a and b= 2 , ab(1) = 4

2. a -> 1+1, ab(2) =1

3. b -> 1+1, ab(2) =1

4. 4+1+1=6

2. (4)

1. a(1) = 3, b(1) = 1, ab(1) = 3

2. a(1) -> a(2) = 2, b(2) =1, ab(2) = 2

3. a(2) -> a(3) = 1, b(3) =1, ab(3) = 1

4. 3+2+1 = 6

3. 4(4-1)/2= 6

Can I do this or do I have to write some proof for it always being equal?