How many solutions does the following congruence have 51x 3

How many solutions does the following congruence have? 51x = 34 mod 646 Consider two solutions the same if they are congruent modulo 646. You can check your answer with a computer if you want, but we\'re looking for a mathematical solution.

Solution

51x 34(mod 646).

GCD(51,646)= 17

Now 34/17=2,Since 17 is a factor of 34, Given Congruence will have 17 solutions.

so 51x 34(mod 646)

==.> 51x=34+646q

==> 3x=2+38q

==> 3x 2(mod 38) ------------ Eq 1

Now we have to find mutiplicative inverse of 3 mod 38 such that xx^-1=1 mod(38)

13*3=39=1mod(38)

So multiply Eq 1 with 13

==> 39x 26(mod 38)

==> x -12(mod 38)

==> x (-12+38)(mod 38)

==> x 26(mod 38)

So the Solution is x= 26+38k

k=0 to 16

Solution: x= 26,64,102,140,178,216,254,292,330,368,406,444,482,520,558,596,634

If x, y are solutions, x>y then x-y=38n (n>=1)