Problem
NT-B2-M11-P005 An Even-Length Palindrome
#5
★★☆☆☆ Level 2 of 5
Prove that every decimal palindrome with an even number of digits is divisible by \(11\).
1001 Problems in Classical Number Theory (method inspiration) · Problem 230
Use the divisibility rule for \(11\) and group symmetric digits.
Let the palindrome have \(2r\) digits: \(a_{2r-1}\ldots a_1a_0\), with \(a_i=a_{2r-1-i}\). In the alternating sum for divisibility by \(11\), symmetric digits have opposite signs because their positions have opposite parity. Hence each pair cancels, the whole sum is \(0\), and the number is divisible by \(11\).
This is short, but it connects symmetry of notation with modular reasoning.