assig9 - over How many troops survived the battle 4(a Find...

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MATH 135 Algebra, Assignment 9 Not to be handed in 1: Solve the following pairs of congruences. (a) x 5 (mod 7) x 8 (mod 15) (b) x 45 (mod 84) x 61 (mod 115) 2: Solve the following pairs of congruences. (a) 15 x 4 (mod 26) 24 x 6 (mod 63) (b) 2 x 3 7 (mod 9) x 2 x + 6 (mod 35) 3: Chinese generals used to count their troops by telling them to form groups of some size n , and then counting the number of troops left over. Suppose there were 5000 troops before a battle, and after the battle it was found that when the troops formed groups of 5 there was 1 left over, when they formed groups of 7 there were none left over, when they formed groups of 11 there were 6 left over, and when they formed groups of 12 there were 5 left
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Unformatted text preview: over. How many troops survived the battle? 4: (a) Find φ ( n ) for all integers n with 20 ≤ n ≤ 30. (b) Find all positive integers n such that φ ( n ) = 60. 5: (a) Show that 2 340 ≡ 1 (mod 341). (b) Show that 21 ± ± (4 n 7 + 7 n 3 + 10 n ) for all integers n . (c) Find a positive integer k such that the number 3 k ends with the digits 0001. (d) Let n = p k for some positive integer k where p is prime with p ≡ 3 (mod 4). Show that the congruence x 2 ≡ -1 (mod n ) has no solution....
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This note was uploaded on 05/04/2010 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.

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