I am having trouble in finding the remainder of $(1\cdot2\cdots102)^3\mod 105$

It is not possible to apply Wilson's Theorem here because 105 is composite. Can anybody help me?


$105 = 3 \cdot 5 \cdot 7, 102!$ is divisible all of $3,5$ and $7,$ therefore $102!^3$ is divisible $105$


Hint: Yes, $105$ is composite. So while you can't use Wilson's theorem, it actually makes the problem a lot easier, rather than more difficult. Think about exactly why Wilson's theorem fails for composite numbers: What is $(n-1)!\pmod n$ in those cases?

  • $\begingroup$ I thought of it but I don't know how to follow $\endgroup$ – Luis Gimeno Sotelo May 20 at 14:58
  • 2
    $\begingroup$ @LuisGimenoSotelo Another hint: $105 = 3\cdot 5\cdot 7$. Now take a close look at your product. What numbers are included there? $\endgroup$ – Arthur May 20 at 14:59
  • $\begingroup$ It is ((n-3)!) ^3. But I have tried some examples of composite numbers and I have found no pattern. Is there a pattern? $\endgroup$ – Luis Gimeno Sotelo May 20 at 15:07
  • $\begingroup$ @LuisGimenoSotelo Don't write it as $(n-3)!^3$, write it as $1\cdot 1\cdot 1\cdot 2\cdot 2\cdot 2\cdot 3\cdots$ (don't write all of it, just imagine that you are). Compare what you get there to $105 = 3\cdot 5\cdot 7$ and see if you can spot anything. Apart from that, I have given you two substantial hints, and Reinstein has given you the actual answer. I don't know what more to do for you. $\endgroup$ – Arthur May 20 at 15:10
  • $\begingroup$ It was very easy when you find that 102! is congruent to 0 (mod 3), to 0 (mod 5) and to 0 (mod 7) and because these numbers are prime to each other, 102! is congruent to 0 (mod 3*5*7). Thank you very much! $\endgroup$ – Luis Gimeno Sotelo May 20 at 15:27

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