Problem
COM-B1-M12-P013 Set 4. Divisible Sum
#13
★★★☆☆ Level 3 of 5
Prove that among any \(8\) integers, one can choose several consecutive numbers whose sum is divisible by \(8\).
Consider partial sums modulo \(8\).
Let \(S_k=a_1+\cdots+a_k\). If some \(S_k\) is divisible by \(8\), done. Otherwise the \(8\) partial sums have only \(7\) nonzero residues, so two have the same residue. Their difference is the sum of a consecutive block and is divisible by \(8\).
Pigeonhole with partial sums.