Two theorems on the addition of residue classes (Q914720)

It is well known that if \(a_ 1,...,a_ m\) are residues modulo n and \(m\geq n\) then some sum \(a_{i_ 1}+...+a_{i_ k}\), \(i_ 1<...<i_ k\), is 0(mod n). The author generalizes such results and proves three theorems on divisibility properties of subsequences of a finite sequence. Let \(f(a_ 1,...,a_ m;j)\) denote the number of sums formed from the \(a_ i\) which are congruent to j (mod n). Two consequences of these theorems are: (1) If \(f(a_ 1,...,a_ m;0)<2^{m-1}\) then \(f(a_ 1,...,a_ m;0)\leq 3\cdot 2^{m-3}\) and 2) Let \(m\geq 2\). There exist \(a_ 1,...,a_ m\) for which \(f(a_ 1,...,a_ m;j)\) is odd if and only if n is not a power of 2.