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Which of the following statements are true?

  1. The 9th power of any positive integer is of the form $19 m $ or $19 m \pm 1$.

  2. For any positive integer $n$, the number $n^{13} - n$ is divisible by 2730.

  3. The number $18! + 1$ is divisible by 437.

I am stuck on this problem. Can anyone help me please...

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Little Fermat, congruences and $(x^{18}-1)=(x^9-1)(x^9+1)$ will get you started. –  Jyrki Lahtonen Jan 5 '13 at 8:38
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1 Answer

Hints:

  • Euler's theorem: for $a$ and $n$ relatively prime, $a^{\varphi(n)}\equiv 1\bmod n$.
  • Wilson's theorem: for a prime number $p$, $(p-1)!\equiv -1\bmod p$.
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how can I aaply these theorems? –  Prasanta Jan 5 '13 at 8:43
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That is for you to think about, for more than 4 minutes at any rate. How do you expect to get better at solving problems if you are not practicing solving problems? –  Zev Chonoles Jan 5 '13 at 8:45
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