Question

4.39 Exercise. Find all pairs of numbers a and b in {2,3,...,11) such that ab = 1 (mod 13). The preceding theorems and examples are giving us a perspective about numbers and their multiplicative inverses modulo a prime p. One conse- quence of this picture is that when we multiply all the numbers from 2 up to (p - 2), we get a number congruent to 1 modulo the prime p.

Answer 1

ab= 1 (mad 13) means 13 divides at-1 aand b a=2,627; a=7,6=2 a=3,6 = 9; a=9, b=3 Tab 14 14 31 (mod 13) 1h = 2x7=742 27 | 27 = 1 (mod 13) 27= 3x9 = 9x3 40 40=1(mad 13) 4024410 = 10x4 = 5x8=845 53 53 2 1 (mad 13) a=4, 6=10; a=10,6=4; a=5,6=8 ; a=8,6=5; :66 166² 1 (mod 13) a=6, b=11; a=11, 6=6 166=6x11=1186 92922 1 (mad13) 1105 105 ²1 (mod 13) 1118 1183 1 (mod 13) 131131² 1 (mod 13) all pair of and bin {2,3,....lly are (2,7), (7,2). (9,9),C9,3), (4,10), (10,4), (5,8),(3,5),(6,11),(,6)