Decimal Representation – 2^m Starting with Specific Digits

Question

Given a finite sequence of decimal digits $a_1,a_2,…,a_n$ prove that there exists a natural number $m$ such that decimal representation of $2^m$ starts with that sequence of digits.

Thanks for your help 🙂

Answer 1

Hint: $\log 2$ is irrational. What can you say about $ \{ n log 2 \}$, the fractional parts?