Homework 3 Solutions on Number Theory - MAS 5215/001...

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MAS 5215/001, Spring 2015CRN: 14560Assignment 3(Solution)Question 1.(A divisibility test.)(2 points)Letn= 1000x+ 100y+ 10z+w.(a) Find an inverse of 1000 modulo 31.(b) Prove that 31|niff 31|(x3y+ 9z+ 4w).(c) Letn= 17,612,242,776. Check if 31 dividesn.(d) Find the remaindernmod 31.Solution.
Question 2.(Exercise 2.36, generalized. The CRT.)(2 points)Letmbe a positive even integer.(a) Prove thatm+ 1,m,m1 are pairwise relatively prime integers.(b) Solve the system of congruences:x2(modm+ 1)x1(modm)x0(modm1).
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