2013 AMC 10A Problem 13

Note that for the number to not be divisible by $5,$ the units digits cannot be either $0$ or $5.$ Let $x$ be the hundreds and units digit and $y$ be the tens digit. Then we want $$2x + y \lt 20.$$ Casing on the $8$ possible values of $x,$ we get: If $x$ is $1, 2, 3,$ or $4,$ then $y$ can be anything since $y \lt 10.$ If $x = 6,$ then $y \lt 8,$ which gives us $8$ solutions. If $x = 7,$ then $y \lt 6,$ which gives us $6$ solutions. If $x = 8,$ then $y \lt 4,$ which gives us $4$ solutions. If $x = 9,$ then $y \lt 2,$ which gives us $2$ solutions. This gives us a total of $$4 \cdot 10 + 8 + 6 + 4 + 2 = 60$$ solutions. Thus, B is the correct answer.