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homework4

# homework4 - multiples of 5 6 Find the least positive...

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THEORY OF NUMBERS, Math 115 A Homework 4 Due Monday October 29 1. A student returning from Europe changes his French and Swiss francs into US. money. He only brought “whole francs”, not fractions of them. He received 19 cents for each French and 59 cents for each Swiss franc, and the total was \$17.06. How much of each type of currency did he exchange? 2. Find all integer solutions of the linear diophantine equations a) 2 x 1 + 5 x 2 = 5 b) 15 x 1 + 6 x 2 = 1 3. The post office of Davis is left with stamps of only two values. They discover that there are only 33 postage amounts that cannot be made up using these stamps. One of them is 46 cents. What are the values of the remaining stamps? 4. For which positive integers m is each of the following true? a) 27 5 (mod m ), b) 1000 1 (mod m ), c) 1331 0 (mod m ), 5. Give a complete system of resisues modulo 7 that consists entirely of
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Unformatted text preview: multiples of 5. 6. Find the least positive residues (that is, the remainders) modulo 47 of each of the following integers: a) 2 32 , b) 2 47 , c) 2 200 , 7. Let a = 4444 4444 . Let b be the sum of the digits of a , when written in decimal. Let c be the sum of the digits of b , when written in decimal. Find the sum of the digits of c . 8. Find all solutions of the following linear congruences: a) 17 x 14 (mod 21), b) 128 x 833 (mod 1001), c) 987 x 610 (mod 1597). 9. For which integers c , 0 c 30, does the congruence 12 x c (mod 30) have solutions? When there are solutions, how many incongruent ones are there? 10. Determine which integers a , where 1 a 14, have an inverse modulo 14. Find the inverse of each. 1...
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