\[\boxed{\text{223}\text{\ (223)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[a > 0;\ \ b > 0.\]
\[a + b = 40\]
\[a = 40 - b\]
\[a \cdot b = b(40 - b) = - b^{2} + 40b =\]
\[= - \left( b^{2} - 2 \cdot 20b + 400 - 400 \right) =\]
\[= - \left( (b - 20)^{2} - 400 \right) =\]
\[= 400 - (b - 20)^{2} \leq 400.\]
\[Наибольшее\ значение\ 400\ \]
\[при\ \ (b - 20)^{2} = 0:\]
\[b - 20 = 0\]
\[b = 20.\]
\[\Longrightarrow a = 40 - b = 40 - 20 = 20.\]
\[Ответ:при\ a = 20;b = 20.\]