Data Sufficiency

I have a question about the solution at the link in the subject line. My ans is (E). If we just use a solution without picking numbers ( picking numbers is tricky and error prone) ... Here is what i get. Please let me know why this is incorrect/correct or please suggest and easier alternative without picking numbers- one that takes lesser time.

What is the remainder when the positive integer x is divided by 8?
(1) When x is divided by 12, the remainder is 5.
(2) When x is divided by 18, the remainder is 11.

SOLUTION)

Consider (1) and the question stem:

x = 8a + r ( where "a" is some integer - We have to find "r")
x = 12b + 5 ( where "b" is some integer)

Subtracting the two equations:

0 = 12b - 8a + 5 - r
=> r-5 = 4(3b-2a)
=> r = 4(3b-2a) + 5
=> r = 4(3b-2a+1) + 1

Deduce that "r" is a multiple of 4 plus 1, i.e. when r is divided by 4 we get a remainder of 1.

We don't get an exact value of "r". So (1) is insufficient.

Consider (b)

(2)

x = 8a + r ( where "a" is some integer - We have to find "r")
x = 18c + 11 ( where "c" is some integer)

Subtracting:
0 = 18c - 8a + 11 - r
=> r - 11 = 18c -8a
=> r = 18c -8a + 11
=> r = 2(9c-4a+5) + 1

This implies "r" is a multiple of 2 plus 1. Again we dont get an exact value of "r"

Combine (1) and (2)

From (1) - r is a multiple of 4 plus 1.
From (2) - r is a multiple of 2 plus 1.

If "r" divided by 4 leaves a remainder of 1. Then "r" divided by 2 will also leave a remainder of 1.

Consider the opposite - if "r" divided by 2 leaves a remainder of 1, then it is not necessary that "r" divided by 4 will also leave a remainder of 1.

So together NOT sufficient

Hence (E).