Here is something taken from what I wrote in my notes on my phone a long time ago...
----------
a + b = 10a + b Divide both sides by a
a/a + b/a = 10a/a + b/a
1 + b/a = 10 + b/a Subtract b/a from both sides
1 = 10
So you'd be like, "NO SOLUTION!!!" right??? WRONG!! because a = 0 IS a solution to the original equation, but when you divided both sides by a it made it look like there was no solution. So that is why you gotta always say a ≠ 0 whenever you divide by an unknown number. So then the final answer would be basically saying "No solution when a ≠ 0. So a = 0 might be a solution might not be I dunno." Then you gotta check if a = 0 is a solution.
----------
So that is why 0 is missing from your list.
When you divided both sides by b, you didn't note that the following equation holds only when b ≠ 0
Otherwise, it is true that ab = 36
There are three solutions to this system of equations (see here) and in both cases where b ≠ 0, ab = 36
But ab = 36 is not the full story! There are more restrictions...
Notice that (1, 36) can't be a solution for (a, b) because
a + ab2 must equal 40b but 1 + 362 ≠ 40(36)
So (1, 36) can't be a solution.
There are more restrictions on a and b besides just ab = 36 ( when b ≠ 0 )