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only one (Posted on 2010-12-28) Difficulty: 1 of 5
There is only one palindromic prime with an even number of digits.

Prove it.

See The Solution Submitted by Ady TZIDON    
Rating: 4.5000 (2 votes)

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Solution Solution (spoiler) | Comment 1 of 3
If a palindrome has an even number of digits, then the sum of the odd digits = the sum of the even digits.   This means that it equals 0 mod 11, because 10^n is 1 mod 11 if n is even and -1 mod 11 if n is odd.

For instance 3223 mod 11 = 
   (3 * 1000 mod 11) + (2 * 100 mod 11) + (2 * 10 mod 11) + 3 =
   3*(-1) + 2*1 + 2*(-1) + 3*1  = 5*(-1) + 5*1 = 0 

Because all palindromes with even digits are divisible by 11, the only one which can be prime is 11.

Edited on December 28, 2010, 6:06 pm

Edited on December 29, 2010, 12:35 am
  Posted by Steve Herman on 2010-12-28 14:55:23

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