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Let $p$ be a prime and let $a$ be an integer not divisible by $p$. Then $$a\times2a\times3a\times...\times(p-1)a\equiv 1\times2\times3\times...\times(p-1) (\bmod p)$$

How do you approach this proof?

Jordan L
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    Do you know Fermat's Little Theorem? – Joshua Pepper Mar 04 '14 at 00:25
  • Not yet. I know this is the theorem that is used to help prove Fermat's Little Theorem though. – Jordan L Mar 04 '14 at 00:26
  • see this http://math.stackexchange.com/questions/698275/prove-if-a-p-1-then-a-2a-3a-pa-is-a-complete-residue-system-modul – r9m Mar 04 '14 at 00:27
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    @Joshua Pepper: That would do it, but the lemma of the OP is intended as the main step in a proof of the Fermat Theorem. – André Nicolas Mar 04 '14 at 00:27
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    Ah, I see. In that case, OP, this result is going to follow from your earlier question - the observation that ${a, 2a, 3a, \ldots, (p-1)a}$ is a rearrangement of ${1,2,3,\ldots,(p-1)}$. – Joshua Pepper Mar 04 '14 at 00:31

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The question you asked recently asks one to show that $a,2a,3a,\dots,pa$ is a complete residue system modulo $p$. If we remove the $pa$, we get a reduced residue system. The numbers $a,2a,3a,\dots,(p-1)a$ are congruent, in some order, to $1,2,3,\dots,p-1$ modulo $p$. It follows that their product $(a)(2a)\cdots((p-1)a)$ is congruent to $(1)(2)\cdots(p-1)$ modulo $p$.

Concrete example: Take $p=7$ and $a=2$. We look at $2,2\cdot 2, 3\cdot 2, \dots, 6\cdot 2$. These are congruent, modulo $7$, to $2,4,6,1,3,5$. These are just $1,2,3,4,5,6$ rearranged. So their product is congruent to $(1)(2)(3)(4)(5)(6)$ modulo $7$.

Do in detail another concrete example, such as $a=4$ and $p=11$, and you will fully understand what's happening.

Community
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André Nicolas
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  • Why is necessary to remove $pa$ to get a reduced residue system? – Jordan L Mar 04 '14 at 00:39
  • $pa$ is congruent to $0$ modulo $p$. A reduced residue system modulo $p$ consists of any set $S$ of $p-1$ numbers, such that any of $,2,3,\dots,p-1$ is congruent to exactly one element of $S$. – André Nicolas Mar 04 '14 at 00:43