Can Sub Lengths Be Equally Divided Among Three TAs?

[remaan]

Hard problem - dividing the subs !

Homework Statement

Anupam brought n > 0 subs, one each of length 1; 2; 3;... ; n, to a grading party.
The three TAs distributed the subs among themselves such that no sub was broken, and each TA
ended up with an equal total length. For what values of n is such a division possible?

Homework Equations

At some point, we may use the the sum formula : n(n+1)/2

The Attempt at a Solution

I tried finding a pattern

n = 1
we have only one sub, doesn't work

n= 2
doesn't work

n=3
doesn't work, as we can't divide this by 3 people.

n=4
doesn't work,

n= 5
it works !
we have 1,2,3,4,5
we can divide by 3 as follows :
one of the TAs will take 5
The other will take 4,1
The third will take 2,3

So, how should I precede with that ?

Do you think this is the right thing ?

 

[TachyonRunner]

It's good that you looked for a pattern right off the start, but I think the main focus is in the formula they gave you (n(n+1)/2). This formula is the what you would use to find 1 + 2 + 3 +...+n. So in the context of this question (n(n + 1))/2 gives you the total length of bread that will be available.
Hope this helps.

 

[remaan]

Mmm...
Ya But the problem with that is:
Knowing how long bread I have Doesn't solve the problem, since
I Am not Able to break the breads apart.

So, any extra hints ??

 

[TachyonRunner]

Think about each bread as an integer. The fact that we can't break apart any bread when we divide the total length by 3 is important as it tells us something about the expression:
$$\frac{1}{3}$$ $$\frac{n(n+1)}{2}$$ , mainly that it can only take on values from a specific set.
Hope this helps

 

[remaan]

Ok. now suppose I found the two numbers -
are there any hints of how to prove them ?