Has this basic number theory conjecture been discovered or proved before? So I was sitting in a car bored for two hours playing around with the calculator on my phone, and I discovered that by picking two numbers (at least one being odd and both being not divisible by each other) I could always add them together to make many different numbers. Furthermore, I could always add and subtract them together make 1; this is significant because I could then negate and indefinitely duplicate the series to create any positive and negative integers. Here are some examples:
8-7=1
16+16+16+16-9-9-9-9-9-9-9=1
17+17+17-5-5-5-5-5-5-5-5-5-5=1

and so on.
For convenience, I preferred to write these equations in the form a*x-b*z=1
 8*1-7*1   = 1
16*4-9*7   = 1
17*3-5*10  = 1

Here is my Conjecture:

Any two non-zero integers indivisible of each other, if at least one
  of them is odd, can be added or subtracted together indefinitely to
  make any integer.

Another way of writing it (though it is more limiting) is this:

Given that a and b are two non-zero integers indivisible of each other, if at least one of them is odd, there always exists some value of x and z for any given integer n that makes the equation ax-bz=n true.

My question is, has anyone ever discovered or proved this conjecture before, and how can it be proven? It almost feels like the solution should be as obvious as the equation a*x-b*z=0 (which just requires finding the lcm of the two numbers) and that it's just a simple quirk in numbers based upon the already laid down basic laws of mathematics.

This is false: Try $15$ and $21$. Note that $3\mid15x-21y$ for all $x,y$. 
However, the rule you are thinking of is basically Bezout's Identity.
