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Unformatted text preview: ICS 6D ( Discrete Mathematic): Solution to Selected Problems on Homework #3 Section 3.4 2. 1  a because there exists integer n , namely n = a , s.t. a = n 1. a  0 because there exists integer n , namely n = 0, s.t. 0 = n a . 6. By assumptions that a  c and b  d , there exist integers n 1 , n 2 s.t. c = n 1 a and d = n 2 b . Therefore cd = ( n 1 n 2 ) ab , i.e. there exists integer n , namely n = n 1 n 2 , s.t. cd = n ( ab ), and therefore ab  cd . 8. This is not true. For example 8  (4 2) but its not true that either 8  4 or 8  2. 12. Note that we showed the opposite direction in class! This direction appears a bit more complicated. Lets show this by arguing the counterpositive, i.e. that if ( a mod n ) negationslash = ( b mod n ) then a negationslash b (mod n ). By the division algorithm theorem there exist unique remainders r 1 and r 2 in Z N s.t. n  ( a r 1 ) and n  ( b r 2 ). (The theorem also says that the quotients are unique, but we dont need it here.) Let)....
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This homework help was uploaded on 01/30/2008 for the course ICS 6D taught by Professor Jarecki during the Fall '07 term at UC Irvine.
 Fall '07
 Jarecki

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