Question

I need help with number 3 on my number theory hw.  

Exercise 1. Figure out how many solutions x2 = x (mod n) has for n = 5,6,7, and then compute how many solutions there are modulo 210. Exercise 2. (a) Find all solutions to x2 +8 = 0 (mod 11). (b) Using your answer to part (a) and Hensel's Lemma, find all solutions to x2 +8 = 0 (mod 121). Exercise 3. Solve f(x) = x3 – x2 + 2x + 1 = 0 (mod 5°) for e = 1,2,3. Exercise 4. Use summation to prove the following identity that we used in the proof of Lagrange's Theorem for Polynomials: 20* – bk = (x – b)(2k-1 + 2k-16+ ... + xbk-2 + 5k-1), for k e Z* Exercise 5 (Optional Challenge). (a) Let p be an odd prime. Show that x2 = 1 (mod p') has only the two solutions x = +1 (mod pl). (b) Let p and q be distinct odd primes, i.e., p + q. Let n = pq. Show that the complete set congruence classes of solutions to x2 = 1 (mod n) is S = {[1], [-1], [py - qz], [qz - py]}, where (Y,Z) is any integer solution to py+qz = 1.

Answer 1

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