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108年 - 108 國立中山大學_碩士班招生考試_資工系(資安)：離散數學與演算法#105778
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(b) Buler's Theorem. For each n Ezt,n ＞ 1, and each a EZ, prove that if gcd(a,n) = 1, then (a中(n) = 1(mod n).
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4. (a) Consider an chessboard. It contains eighty-one squares and one square. How many squares?
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(b) Now consider an chessboard for some fixed . ForI , how many squares are contained in this chessboard?
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5. Let be a set of five positive integers the maximum of which is at most 9. Prove that the sums of the elements in all the nonempty subsets of S cannot all be distinct.
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6. (a) Fermat's Theorem. If is a prime, prove that for each .
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(a).
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(b)
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(a) 251,1920
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(b) 1371,2587
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(c) 1689,6001
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(a) Apply Dijkstra's algorithm to the graph shown in the following Fig. 1 and determine the shortest distance from vertex b to each of the other vertices in the graph.
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