How many palindromes below 10000 (i.e. between 0 and 9999 inclusively) are not divisible by 11?
List them.
11 will divide all palindromes with an even number of digits, as the evenposition digits will add up to the same as the oddposition digits so the difference between these sums is zero.
None of the singledigit integers is divisible by 11, except if you consider zero.
The threedigit numbers will be divisible by 11 only if the middle digit is twice the first digit (that is, the middle digit equals the sum of the first and last digits) or the two outer digits add to 11 more than the middle digit.
So the indivisible list is:
1, 2, 3, 4, 5, 6, 7, 8, 9,
101, 111, 131, 141, 151, 161, 171, 181, 191,
202, 212, 222, 232, 252, 262, 272, 282, 292,
303, 313, 323, 333, 343, 353, 373, 383, 393,
404, 414, 424, 434, 444, 454, 464, 474, 494,
505, 515, 525, 535, 545, 555, 565, 575, 585, 595,
606, 626, 636, 646, 656, 666, 676, 686, 696,
707, 717, 727, 747, 757, 767, 777, 787, 797,
808, 818, 828, 838, 848, 868, 878, 888, 898,
909, 919, 929, 939, 949, 959, 969, 989, 999

Posted by Charlie
on 20121226 17:32:47 