I know that (x*y)%p = ((x%p) * (y%p))%p holds true for a prime p. Is this equation valid when p is a composite number? How do we write this equation when p is a composite number?

  • 1
    $\begingroup$ Of course it is $\endgroup$
    – gukoff
    Jan 3 '14 at 20:23
  • 3
    $\begingroup$ It is valid. No change required, except for clarity to replace $p$ by $m$. $\endgroup$ Jan 3 '14 at 20:29
  • $\begingroup$ Perhaps it is just my browser but just before each instance of $p$ I see the character "%". I am not familiar with this notation. $\endgroup$
    – Jay
    Jan 3 '14 at 20:48

The map $\mathbb Z\to \mathbb Z/(n\mathbb Z)$ is a ring homomorphism; this means the operations of addition, subtraction, and multiplication in $\mathbb Z/(n\mathbb Z)$ behave in the expected way.

The unusual feature of multiplication in $\mathbb Z/(n\mathbb Z)$ when $n$ is composite is that the product of two nonzero elements can be zero, as in $2\cdot 3 \equiv 0 \mod 6$.


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