## Counting Digits

#### Problem

How many digits does the number 2^{1000} contain?

#### Solution

As 2^{1000} is not a multiple of 10, it follows that,

10^{m} 2^{1000} 10^{m + 1}, where 10^{m} contains `m` + 1 digits.

Solving 2^{1000} = 10^{k}, where `m` `k` `m` + 1

`k` = `log` 2^{1000} = 1000 `log` 2 301.02999... , so `m` = [1000 `log` 2] = 301.

Hence 2^{1000} contains 302 digits.

What is the least value of `n`, such that 2^{n} contains exactly one million digits?

Problem ID: 128 (Oct 2003) Difficulty: 4 Star