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I roughly understand modular arithmetic but I am having trouble starting the problem. I can prove it for just integers but I can't seem to relate it to mod primes and composites?

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    $\begingroup$ By definition, a composite number $q$ is the product of two smaller integers. What is that product modulo $q$? $\endgroup$ – eyeballfrog Sep 27 at 7:15
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Here are two strategies that will work:

  1. Start with a single example. Is the statement true in the integers modulo $6$? Why not, exactly? How does your argument generalize to other composite numbers?
  2. Start by writing out definitions. What does it mean for a modulus $n$ to be a composite number? What does it mean for an integer to be congruent to $0$ modulo $n$? Can you find a way to plug these definitions into the problem you're trying to solve?
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Well, in the residue class ring ${\Bbb Z}_m$, $m\geq 2$, each element $a\ne 0$ is either a unit or a zero divisor. An element $a\ne 0$ is a unit if $\gcd(a,m)=1$, otherwise a zero divisor.

The implication $ab=0\rightarrow a=0\vee b=0$ holds iff $a$ or $b$ is not a zero divisor.

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