Problem
NT-B1-M10-P013 Even-Length Palindrome
#13
★★★☆☆ Level 3 of 5
Prove that every decimal palindrome with an even number of digits is divisible by \(11\).
In the alternating sum, symmetric digits have opposite signs.
Because the number of digits is even, each pair of equal symmetric digits enters the alternating sum with opposite signs. Thus the alternating sum is \(0\), so the number is divisible by \(11\).
Classic olympiad mini-lemma.