Prove that it is no possible to separate rocks weighing $1,3,...33$ pounds into any number of piles so that the weight of each pile is the same.

Question

I tries to solve it by attempting to show that the weights cannot be equal by showing through pigeonhole, but I got stuck since there are too many cases to prove. How would I solve this?

Answer 1

The total weight of all the rocks is $289$ which is prime. If you separate the rocks in to $a$ piles each weighing $b$ pounds, then $ab = 289$.

Edit: Sorry, $289 = 17^2$, so $ab =17^2$. So the only possibility is $a=b=17$ which is easy to eliminate.