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Counting Digits

Problem

How many digits does the number 21000 contain?


Solution

As 21000 is not a multiple of 10, it follows that,
10m less than 21000 less than 10m + 1, where 10m contains m + 1 digits.

Solving 21000 = 10k, where m less than k less than m + 1

k = log 21000 = 1000 log 2 approximately 301.02999... , so m = [1000 log 2] = 301.

Hence 21000 contains 302 digits.

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

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

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