I have information about Euler theorem: $a^{p}=a\mod{p}$ and Wilson's theorem: $(p-1)!+1=0\mod{p}$. But I have no clue how to approach this one. Please give some hints.
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2Well, those are the tools. What is $22!\pmod {23}$ for example,? – lulu Oct 01 '22 at 18:11
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2Knowing $23=22+1$ and $44= 2 \times 22$ may help – Henry Oct 01 '22 at 18:11
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1$!\bmod 23!:\ \color{#c00}{22!\equiv -1},$ by Wilson, and Fermat $\Rightarrow 6^{22}\equiv 1\Rightarrow \color{#0a0}{6^{44} = (6^{22})^2\equiv 1^2\equiv 1},,$ so $,\color{#0a0}{6^{44}},\color{#c00}{22!}+3\equiv \color{#0a0}1,(\color{#c00}{-1})+3\equiv 2,$ by Congruence Sum & Product Rules – Bill Dubuque Oct 01 '22 at 18:50