Question

8-7. Find the smallest positive integer a such that 5:13 +13n" + a(9) = 0 (mod 65) for all integers n.

Answer 1

Ans: 51² +13n5+ a (9n) = 0 (mod 65) Remember lil fermat can be written as Vn, nan mods and Vn, n3 n mod 13. So we have to solve the system of simultaneous congruences: $13n+qan= 3n-an zo mod 5 1 sn + qan= 5n-4 an> 0 mod 13 a = 3 mod 5 4a = 5 ) a = -3.5. =-2 mod13 Now a Bezout's relation between 5 and 13 is 2.13-5.51 so solution of the system of congruences is n=3.2.13-(-2) 5.5 = 63 mod 65 Thus the smallest positive integer a satisfying the given congruence for all in is 63. og 3