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Computer Science 70 - Fall 2000 - Russell - Midterm 2

# Computer Science 70 - Fall 2000 - Russell - Midterm 2 - CS...

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CS 70, Midterm #2, Fall 2000 CS 70, Fall 2000 Midterm 2 Papadimtriou/Russell/Sinclair Problem #1 (20 pts.) Extended GCD For each of the following equations, find a pair of integers x and y that satisfies that equation or prove that no such pair exists. [ Note : these are not simultaneous equations.] (a) (5 pts) 13 x + 21 y = 1 (b) (5 pts) 13 x + 21 y = 2 (c) (5 pts) 33 x + 21 y = 1 (d) (5 pts) 33 x + 21 y = 3 Problem #2 (20+5 pts.) Perfect Squares Let p be a prime greater than 2. An integer y is called a perfect square modulo p if y = x ² mod p for some integer x ; x is called a square root of y modulo p . (a) (6 pts) Which among the integers 0,1,...,10 are perfect squares modulo 11? (b) (8 pts) Prove rigorously that each integer y , where 0 < y < p , has either zero or two square roots modulo p . [ Hint : Suppose w and x are square roots of y ; what can you deduce about the relationship between them?] (c) (6 pts) Using the result in (b), prove that there are exactly ( p +1)/2 perfect squares modulo p . (d) (5 pts, extra credit) Prove that there are at least p /3 perfect cubes modulo p .

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Computer Science 70 - Fall 2000 - Russell - Midterm 2 - CS...

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