Problem
COM-B1-M05-P012 Block with Sum Divisible by \(10\)
#12
★★☆☆☆ Level 2 of 5
Prove that among any \(10\) integers, there is a nonempty consecutive block whose sum is divisible by \(10\).
Consider partial sums.
Let \(s_i\) be the sum of the first \(i\) numbers. If some \(s_i\) is divisible by \(10\), done. Otherwise the \(10\) partial sums have only \(9\) nonzero residues, so two have the same residue. Their difference is the sum of a consecutive block divisible by \(10\).
Partial sums as boxes.