Project Euler Problem 16 in F#

It's time for another Project Euler exercise. This time around it is problem 16:

215 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.

What is the sum of the digits of the number 21000?

Analysis

I have tried to find an elegant way to solve this problem at a mathematical level, but I couldn't gain any insight relating powers of 2 to their representation in base 10 (I'd love to know if such a thing exists). Therefore I resorted to solve this problem with brute force.

Luckily F# (and now all other .NET 4.0 cousin languages) sports the BigInteger numeric type which helps tackling such huge numbers.

There are basically two approaches to solve this problem:

• Transform the result of exponentiation into a string and iterate through the string adding up all the individual digits, or
• Do without strings, and decompose the big number into the individual digits on my own.

I have decided to take the second route because it seemed like a more interesting exercise (the first approach could probably lead to a one-liner).

Let's consider a generic number $$n$$ and consider how we can describe it as a sum of powers of 10:

$$n = \sum_{i=0}^k d_i10^i$$

For example

$$423 = 3 \cdot 10^0 + 2\cdot 10^1 + 4 \cdot 10^2$$

and $$k = 2, d_0 = 3, d_1 = 2, d_2 = 4$$

So, given a generic number $$n$$, we need to find the highest power of ten ($$k$$) such that $$\frac{n}{10^k} > 0$$

(logarithms will help us there). After $$k$$ is found, we can start diving $$n$$ by $$10^k$$ to find $$d_k$$ and use the reminder of the division to calculate $$d_{k-1}$$ and so forth until we reach $$d_0$$.

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