Answer
The two lectures will start at the same time at\[11:48\text{ A}\text{.M}\text{.}\]
Work Step by Step
As the length of videotapes are \[42\]and\[56\].
The time at which both videotapes start together will be multiple of both \[42\]and\[56\].
So, in order to find the least time when both tapes start together, find Least Common Multiple(LCM) of \[42\]and\[56\].
Prime factorization of \[42\]and\[56\]:
\[\begin{align}
& 42={{2}^{1}}\times {{3}^{1}}\times {{7}^{1}} \\
& 56={{2}^{3}}\times {{7}^{1}} \\
\end{align}\]
Calculate Least Common Multiple (LCM) of \[42\]and\[56\]:
\[\begin{align}
& \text{LCM}\left( 42,56 \right)={{2}^{3}}\times {{3}^{1}}\times {{7}^{1}} \\
& =168
\end{align}\]
Convert \[168\] minutes to hours:
\[\frac{168}{60}=2+\frac{48}{60}\]
That means,
\[2\text{ hours and }48\min \]
The two lectures will start at the same time at\[9:00\text{ A}\text{.M}\text{. }+2\text{ hours }48\text{ min}=11:48\text{ A}\text{.M}\text{.}\]
Hence, the two lectures will start at the same time at\[11:48\text{ A}\text{.M}\text{.}\]
