Congruence question?

Question

I have to do a bunch of problems on congruence like the following below. I was wondering if anyone could explain how the following is done so that I have an example to look at for the others. Thanks in advance:

Find all incongruent solutions to the following congruence:

6x = 5 (mod 15)

Answer 1

You can't just multiply by the inverse of 6 because15 is not prime and 6 may even not have an inverse. The simplest way to do is to first make a list of numbers in the class of 5 mod 15

5,20,35,50,65,80,95,110,...and then find all multiples of 6 in it.

I think that there is no solution because 6 does not divide 15n+5 because 15n+5=5(3n+1) and 6 doesn't divide 5 but 6 doesn't divide 3n+1 either because 3n+1 will never be a multiple of 3 and in order to be a multiple of 6, the number has to be a multiple of 3.

Another way to look at it is that from the list above, the numbers alternate between odd and even. So our solution would have to be the even one because 6 is even. So the pattern is that you start with 20 and then keep adding 30 to get all the even numbers in the above list. But nothing in that list would be divisible by 6 because 20 is not divisible by 6 and then you are adding 30 which is divisible by 6 so the form is 30n+20 which is not divisible by 6.

Done!

Answer 2

6x = 5 (mod 15)
(6x-5)/15 not = integer is non-congruent solution
So 6x-5 not = 15n
x not = (15n+5)/6 where n = 1,2,3, ....

Answer 3

6x-15y = 5 is an equivalent equation to your congruence
as m.c.d(6,15)=3 does not divide 5 there is no solution.

Answer 4

you multiply with the inverse of 6 modulo 15
oh yes, 6 doesnt have an inverse, i take my words back