# ass5 - (1 12 x = 2 in Z 19 Z(2 7 x = 2 in Z 24 Z(3 31 x = 1...

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ASSIGNMENT 5 - MATH235, FALL 2007 Submit by 16:00, Monday, October 15 (use the designated mailbox in Burnside Hall, 10 th floor). 1. To check if you had multiplied correctly two large numbers A and B , A × B = C , you can make the following check: sum the digits of A ; keep doing it repeatedly until you get a single digit number a . Do the same for B and C and get numbers b, c . If you have multiplied correctly, the sum of digits of ab is c . Prove that this is so. This is called in French “preuve par neuf”. Example: I have multiplied A = 367542 by B = 687653 and got C = 252741358926. To check (though this doesn’t prove the mulitplication is correct) I do: 3 + 6 + 7 + 5 + 4 + 2 = 27 , 2 + 7 = 9 and a = 9. Also 6 + 8 + 7 + 6 + 5 + 3 = 35 , 3 + 5 = 8 and b = 8. ab = 72 and its sum of digits is 9. On the other hand 2 + 5 + 2 + 7 + 4 + 1 + 3 + 5 + 8 + 9 + 2 + 6 = 54 , 5 + 4 = 9. So it checks. 2. (1) Solve that equation x 2 + x = 0 in Z / 5 Z . (2) Solve that equation x 2 + x = 0 in Z / 6 Z . (3) Solve that equation x 2 + x = 0 in Z /p Z , where p is prime. 3. Solve each of the following equations:
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Unformatted text preview: (1) 12 x = 2 in Z / 19 Z . (2) 7 x = 2 in Z / 24 Z . (3) 31 x = 1 in Z / 50 Z . (4) 34 x = 1 in Z / 97 Z . (5) 27 x = 2 in Z / 40 Z . (6) 15 x = 5 in Z / 63 Z . 4. (1) Let p > 2 be a prime. Prove that an equation of the form ax 2 + bx + c (where a,b,c ∈ F p ,a 6 = 0) has a solution in Z /p Z if and only if b 2-4 ac is a square in Z /p Z . If this is so, prove that the solutions are given by the familiar formula. (2) Determine for which values of a the equation x 2 + x + a has a solution in Z / 7 Z . 5. In each case, divide f ( x ) by g ( x ) with residue: (1) f ( x ) = 3 x 4-2 x 3 + 6 x 2-x + 2, g ( x ) = x 2 + x + 1 in Q [ x ]. (2) f ( x ) = x 4-7 x + 1, g ( x ) = 2 x 2 + 1 in Q [ x ]. (3) f ( x ) = 2 x 4 + x 2-x + 1, g ( x ) = 2 x-1 in Z / 5 Z [ x ]. (4) f ( x ) = 4 x 4 + 2 x 3 + 6 x 2 + 4 x + 5, g ( x ) = 3 x 2 + 2 in Z / 7 Z [ x ]....
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