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Properties $(\mod {m})$
09-13-2010, 05:22 PM
Post: #1
Properties $(\mod {m})$
(1) $\gcd(a,m) = 1 \Leftrightarrow \overline{a}$ is invertible in $\mathbb{Z}_m$
(2) $(c \mid ab) \wedge (a,c)=1 \Rightarrow c \mid b$
(3) $((a,m) = (b,m) = 1 )\Rightarrow (ab,m) = 1$
(4) $(ax \equiv ay \mod{m}) \wedge ((a,m) = 1) \Rightarrow (x \equiv y \mod{ m})$
(5) $d = (a,b) \Rightarrow (a/d,b/d) = 1$
(6) $(ax \equiv ay \mod{m}) \wedge (d = (a,m)) \Rightarrow x (\equiv y \mod{m/d})$
(7) $(a_i \mid b, \; i = \overline{1,r}) \wedge ((a_i,a_j) = 1, \; i \ne j) \Rightarrow (\prod_i a_i )\mid b$
(8) $ab = \gcd(a,b) \text{lcm}(a,b)$
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