math 311w homework6

math 311w homework6 - Z 5 . Use it to Solve the following...

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Homework 6. Due Monday,March 22, 2010 Give rigorous solutions of all problems. Problem 1. Prove that if k is even then it is possible to put + and - signs before numbers 1 , 2 ,..., 2 k - 1 , 2 k in such away that the result was zero. For example, - 1 + 2 + 3 - 4 = 0 If k is odd prove that is is impossible. Problem 2. Find the inverse of the numbers [5] 8 , [4] 7 , [31] 72 . Problem 3. Prove that the remainder of the number a 5 a 4 a 3 a 2 a 1 a 0 over 99 is the same as the remainder of the number a 5 a 4 + a 3 a 2 + a 1 a 0 Problem 4. Write the multiplication table for
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Unformatted text preview: Z 5 . Use it to Solve the following system of linear equations: 2 x + 3 y 1 ( mod 5) 3 x-2 y 2 ( mod 5) Problem 5. Solve the game lights out for 2 2 case. Denote the colors of the cells as b 1 ,b 2 ,b 3 ,b 4 , the number of the clicks that you need to make for a particular cell as x 1 ,x 2 ,x 3 ,x 4 , write a system of equations and solve it. Remember that all x and b are from Z 2 . 1...
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