\[\boxed{\mathbf{1.}}\]
\[1)\ \frac{6}{17} \cdot 5 = \frac{6 \cdot 5}{17} = \frac{30}{17} = 1\frac{13}{17}\]
\[2)\ \frac{4}{19} \cdot 8 = \frac{4 \cdot 8}{19} = \frac{32}{19} = 1\frac{13}{19}\]
\[3)\ \ 5\frac{2}{3} \cdot 4 = \frac{17}{3} \cdot 4 = \frac{17 \cdot 4}{3} =\]
\[= \frac{68}{3} = 22\frac{2}{3}\]
\[4)\ \frac{2}{3}\ :7 = \frac{2}{3 \cdot 7} = \frac{2}{21}\]
\[5)\ \frac{16}{5}\ :9 = \frac{16}{5 \cdot 9} = \frac{16}{45}\]
\[\boxed{\mathbf{2.}}\]
\[1)\ \ 2\frac{1}{4} \cdot 11 = \frac{9}{4} \cdot 11 = \frac{9 \cdot 11}{4} =\]
\[= \frac{99}{4} = 24\frac{3}{4}\]
\[2)\ 3\frac{2}{5} \cdot 4 = \frac{17}{5} \cdot 4 = \frac{17 \cdot 4}{5} =\]
\[= \frac{68}{5} = 13\frac{3}{5}\]
\[3)\ 1\frac{5}{7}\ :6 = \frac{12}{7}\ :6 = \frac{12}{7 \cdot 6} = \frac{12}{42}\]
\[4)\ 2\frac{7}{9}\ :11 = \frac{25}{9}\ :11 =\]
\[= \frac{25}{9 \cdot 11} = \frac{25}{99}\]
\[\mathbf{37.\ 1.\ Основное\ свойство\ дроби}\]