## Balanced Numbers

### Problem 217

Published on Friday, 14th November 2008, 09:00 pm; Solved by 945; Difficulty rating: 70%
A positive integer with `k` (decimal) digits is called balanced if its first ⌈^{k}/_{2}⌉ digits sum to the same value as its last ⌈^{k}/_{2}⌉ digits, where ⌈`x`⌉, pronounced ceiling of `x`, is the smallest integer ≥ `x`, thus ⌈π⌉ = 4 and ⌈5⌉ = 5.

So, for example, all palindromes are balanced, as is 13722.

Let T(`n`) be the sum of all balanced numbers less than 10^{n}.

Thus: T(1) = 45, T(2) = 540 and T(5) = 334795890.

Find T(47) mod 3^{15}