\[\boxed{\text{674}\text{\ (674)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[Пусть\ стороны\ треугольника - a_{1},a_{2},a_{3}.\ Тогда\ периметр\ равен:\]
\[P = a_{1} + a_{2} + a_{3} = 24\ см.\]
\[a_{2} = a_{1} + d,\ \ a_{3} = a_{1} + 2d.\]
\[a_{1} + a_{2} + a_{3} = a_{1} + a_{1} + d + a_{1} + 2d = 3a_{1} + 3d = 3 \cdot \left( a_{1} + d \right) = 24;\]
\[a_{1} + d = 8 \Longrightarrow a_{2} = 8;\]
\[a_{1} + 8 + a_{3} = 24 \Longrightarrow a_{1} + a_{3} = 16 \Longrightarrow a_{1} = 16 - a_{3};\]
\[a_{1} + a_{2} > a_{3};\ \ \ a_{1} + 8 > a_{3};\ \ a_{3} + 8 > a_{1} \Longrightarrow по\ свойствам\ треугольника.\]
\[\Longrightarrow a_{1} \in \left\{ 5;6;7;8;9;10;11 \right\},\ \ a_{3} \in \left\{ 5;6;7;8;9;10;11 \right\}.\]