Question

g-2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t1 2 3 4 567 89 10 11 12 13 14 151617 18 g 2 481613714918 17 15 11 36125101 Question 1. (a) Find the least positive residue of 126 modulo 19 (efficiently, using the given information!) (b) Find the number of solutions to 15 = 12 (mod 19) (c) Find all the solutions to r1512 (mod 19) Question 2. Evaluate given for any other method and using the results of the table. No marks will be

Answer 1

@ To find the least positivereaidue 126 ie al solu-ion x such that Now from table emod ulo 1s) ⓑ To find number 여 sduHans to x's-12cm여(9) from table any x e,,scan be writen s we have o nd soluioo et) e12 cmod 1) CS Scanned with

St-1S +he congruence axsbemed m) gcdča,m) | b fna.qt、du.Hons nohw that har soluto n here have α congruence fDm par+ (b) 15と 15 mod I @ we tzr, t=13 thi's congruence holds Thus usin g there values 여 t 'n eqnO ミ 12 cm od 19) Scanned

19 From table 1 9 by table 14 19 din ce C ):1 iff 19 8 f's mot quadratic residue modulo 1g Scanned with CamScanner