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To check that x^ {100} - 1 is divisible by 1000, it will suffice to check that it is divisible by 8 and 125. For each of these, you can use Euler's theorem, which will tell you that x^ {p^ {k-1} (p-1)} \equiv 1 \pmod {p^k} ...

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2018年11月1日 · I wish to show that 222x <100100x for x sufficiently large. I have taken logs (base 10) of both sides to get 2 (2x − 1) log102 and 100x. It is not immediately clear how I can prove that the second term here is larger than the first. Any help appreciated.