N is both a a palindrome and a sum of eleven consecutive positive integers.

How many possible values for N exist below 80,000?

Please provide the lowest and the highest samples.

(In reply to

re(3): Simplifying the question by Ady TZIDON)

ANALYTICAL SOLUTION:

Any multiple of 11 starting with 66 works, if it is a palindrome

Two digits: 66, 77, 88, 99 -- total of 4

Three digits:

mod 11, 0 = aba = 101*a + b*10 = 2a - b

b = 2a mod 11 has single digit solutions for all a except a = 5

Total solutions = 9 - 1 = 8

Four digits:

mod 11, 0 = abba = 1001*a + b*110 = 0

all a and b work

Total solutions = 90 (ie, 10 to 99)

Five digits:

mod 11, 0 = abcba = 10001*a + b*1010 + 100c = 2a - 2b + c

c = 2(b-a) mod 11 has single digit solutions as long as (b-a) <> 5 mod 11

ab under 80 that do not work are 16, 27, 38, 49, 60, 71

Total solutions under 8 = 70 - 6 = 64

Grand total = 4 + 8 + 90 + 64 = 166, which has the advantage of agreeing with Charlie's computer