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
re(4): Simplifying the question
