10. The number of real pairs $(a, b)$ that make the system of equations $\left\{\begin{array}{l}a x+b y=1 \\ x^{2}+y^{2}=50\end{array}\right.$ have solutions, and all solutions are integers, is $\qquad$.
10. 72

Because there are 12 integer points on $x^{2}+y^{2}=50$: $(1,7),(1,-7),(-1,7),(-1,-7)$, $(7,1),(-7,1),(7,-1),(-7,-1),(5,5),(5,-5),(-5,5),(-5,-5)$, each pair of points determines 1 line, totaling $C_{12}^{2}=66$ lines, but 6 of them pass through the origin, and the plane $a x+b y=1$ does not pass through the origin, so there are 60 lines in total; additionally, through each point, 1 tangent line can be drawn, totaling 12 lines, so there are 60 + 12 = 72 lines in total, thus there are 72 corresponding $(a, b)$.