Q . Suppose that we state that a positive integer number is called “special” if the set {1,2,3, . . . ,2016} can be split into subsets, all of them with the same number of elements and the same sum of elements.
a) What are the smallest > 1 and the largest < 2016 “special” numbers? Why?
b) Which of the two numbers 336 and 672, if any, is a “special” number? Why?
Obviously $nn$ must be a divisor of $2016$, and also of $1+2+\ldots+2016 = 2016 \times 2017/2$, thus a divisor of $1008$. So $672$ is not special.
To see that $1008$ is special, split into subsets $\{1,2016\},\; \{2,2015\},\; \ldots,\; \{1008, 1009\}$.
Any divisor $d > 1$ of $1008$ is special: split up the $1008$ pairs above into $d$ sets of $1008/d$ pairs.