MathLabs

Problem 5

List AA contains the decimal numbers 10k10^k for integers k≥1k\ge1. Lists BB and CC contain these same numbers written in bases 22 and 55, respectively. Prove that for every integer n>1n>1, exactly one number in exactly one of BB or CC has exactly nn digits.
Step 2 of 4: Count quinary digits
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ck=⌊klog⁡510⌋+1c_k=\lfloor k\log_5 10\rfloor+1
Detailed analysis

The same argument in base 55 gives ck=⌊klog⁡510⌋+1c_k=\lfloor k\log_5 10\rfloor+1.