Sum of all numbers possible by removing digits of n
Source: Indonesian Mathematics Olympiad 2011, Day 1, Problem 1
September 13, 2011
modular arithmeticnumber theory proposednumber theory
Problem Statement
For a number in base , let be the sum of all numbers possible by removing some digits of (including none and all). For example, if , ; this is formed by taking the sums of all numbers obtained when removing no digit from (1234), removing one digit from (123, 124, 134, 234), removing two digits from (12, 13, 14, 23, 24, 34), removing three digits from (1, 2, 3, 4), and removing all digits from (0). If is a 2011-digit integer, prove that is divisible by .Remark: If a number appears twice or more, it is counted as many times as it appears. For example, with the number , appears three times (by removing the first digit, giving which is equal to , removing the first two digits, or removing the last two digits), so it is counted three times.