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Unformatted text preview: 1.7. MODULAR ARITHMETIC 41 and .OEa C OEb/ C OEc D OEa C .OEb C OEc/: (b) OE0 is an identity element for addition; that is, for all OEa 2 Z n , OE0 C OEa D OEa: (c) Every element OEa of Z n has an additive inverse OE a , satisfying OEa C OE a D OE0: (d) Multiplication on Z n is commutative and associative; that is, for all OEa;OEb;OEc 2 Z n , OEaOEb D OEbOEa; and .OEaOEb/OEc D OEa.OEbOEc/: (e) OE1 is an identity for multiplication; that is, for all OEa 2 Z n , OE1OEa D OEa: (f) The distributive law hold; that is, for all OEa;OEb;OEc 2 Z n , OEa.OEb C OEc/ D OEaOEb C OEaOEc: Multiplication in Z n has features that you might not expect. On the one hand, nonzero elements can sometimes have a zero product. For ex ample, in Z 6 , OE4OE3 D OE12 D OE0 . We call a nonzero element OEa a zero divisor if there exists a nonzero element OEb such that OEaOEb D OE0 . Thus, in Z 6 , OE4 and OE3...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Addition, Multiplication

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