by Tapi
Last Updated May 15, 2019 18:20 PM

Prove that $2^{30}$ has at least two repeated digits.

I assume that the question is asking me to prove that $2^{30}$ has at least one digit that appears twice. Correct me if I'm wrong. (I later checked $2^{30}$ has three digits each of which appears twice, I initially thought that if I could prove the $2^{30}$ has 11 digits, then I can prove the given, but calculated the number of digits only to find out that it has 10 digits).

$2^{30}$ has $10$ decimal digits, as $10^9 < 2^{30} < 10^{10}$. If none were repeated, each of the $10$ digits $0$ to $9$ would appear once. But if that were the case, the sum of digits would be $45$, which would make the number divisible by $9$, and $2^{30}$ is not.

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