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

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).

• Hint: What is $2^{30} \pmod 9$? – user670344 May 15 '19 at 18:14
• $2^{30}=1073741824$, so ... – Sil May 19 '19 at 17:10

$$2^{30}$$ 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.
• This answer seems to depend on convincing yourself that $2^{30}$ has 10 digits without actually multiplying it out or resorting to logarithms (since doing that would violate the spirit of the question). The programmer in me knows that $2^{10}$ is $1024$. So $2^{30} = (2^{10})^3$ is a little more than $1000^3$, which is easy to see has 10 digits. – csd May 15 '19 at 18:57