Question 107
Let $S$ be a positive integer divisible by all the integers $1, 2, \ldots, 2015$ and $a_{1}, a_{2}, \ldots, a_{k}$ numbers in $\{1, 2, \ldots, 2015\}$ such that
$$
2S \leq a_{1} + a_{2} + \ldots + a_{k},
$$
Prove that we can select from $a_{1}, \ldots, a_{k}$ some numbers so that the sum of these selected numbers is equal to $S$.
$$
2S \leq a_{1} + a_{2} + \ldots + a_{k},
$$
Prove that we can select from $a_{1}, \ldots, a_{k}$ some numbers so that the sum of these selected numbers is equal to $S$.