Problem
COM-B1-M05-P014 Sum or Difference Divisible by \(10\)
#14
★★★☆☆ Level 3 of 5
Prove that among any \(7\) integers, two have either sum or difference divisible by \(10\).
Group residues: \(0\), \(5\), \(\{1,9\}\), \(\{2,8\}\), \(\{3,7\}\), \(\{4,6\}\).
There are \(6\) boxes of residues: \(0\), \(5\), and opposite residue pairs. Seven numbers put two in the same box. If their residues are equal, the difference is divisible by \(10\). If opposite, the sum is divisible by \(10\).
Boxes are built for two possible conclusions.