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

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).

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$$2^{30}$$ 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.