Show that if $\gcd(r,s_1) =\gcd(r,s_2) = 1$, then $\gcd(r,s_1s_2) = 1$ Never mind the question. I want to try to solve that on my own. What I want to understand is how this: "Hint. $1 = ar + bs_1,\ 1 = ar + bs_2$" relates to solving it. 
I'm a little confused by this statement, especially since it applies to integers. 
If we say ar + bs = 1, that says to me that you have different multiples of r and s. And when you add those multiples you get 1. How can this happen unless you are adding fraction together or something, or one expression equals 1 while the other equals 0?

Integers can be negative, too. You can have $2 \times 5 + ( -3) \times 3= 1$ for example. 
