In number theory solve the congruency 53x 96 mod 79Solutionc

In number theory:

solve the congruency

53x 96 (mod 79)

Solution

clearly x is of the form c +79 k , where c,k belongs to integer

53x 96 (mod 79) (putting the value of x = c+79k)

53 (c +79 k) 96 (mod 79) 53 *79 k is divisible by 79   

53 c 96 (mod 79) 96 -79 = 17

53 c 17 (mod 79) 17 and 79 are co-prime

53c / 17 1 (mod 79) 53 c = 51c + 2c

(51c +2 c) /17    1 (mod 79) sending 1 on LHS

(3c - 1) +2c/17 0 (mod 79)   2c/17 is integer   ,hence c = 17 a ,putting a = 1,2,3 ... we see a=3 satisfy the solution,

c = 51   

x = 51 + 79 k