Homework 5 on Elementary Number Theory - MAS 4214/001...

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MAS 4214/001 Elementary Number Theory, Fall 2014CRN: 80544Assignment 5Qyestion 1.(2 pts)This problem is about solvingaxb(modm), wherea= 2123 andm= 4632.(a) Use the extended Euclidean Algorithm to findd= gcd(m,a), ands,tso thatd=ms+at.(b) Solve the congruenceaxb(modm), whenb= 579.(c) Solve the congruenceaxb(modm), whenb= 379.Question 2. (2 pts) (Exercise 20.5)Determine whether or not each of the following is true.Give reasonsin each case.(a)x3 (mod 7)gcd(x,7) = 1(b) 12x15 (mod 35)4x5 (mod 7)(c)x6 (mod 12)gcd(x,12) = 6(d) 3x3y(mod 17)xy(mod 17)(e) 5xy(mod 6)15x3y(mod 18)(f) 5xy(mod 6)15x3y(mod 6)(g) 12x12y(mod 15)xy(mod 5)(h)x73 (mod 75)xmod 75 = 73(i)x73 (mod 75) and 0x<75x= 73(j) There is no integerxsuch that 12x7 (mod 33).
.(2 pt2)Solve the following systems of congruences:(a)x6(mod 18)x33(mod 75).(b)x0(mod 2)x6(mod 9)x8(mod 25).
.(1 pt)Solve the following systems of congruences:

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