Answer
\[\text{GCD}=2\text{ and LCM}=27432\]
Work Step by Step
Write \[216\text{ and }254\]in terms of prime factors as follows:
\[\begin{align}
& 216=2\times 2\times 2\times 3\times 3\times 3 \\
& 254=2\times 127 \\
\end{align}\]
Further express the factors in terms of exponents:
\[\begin{align}
& 216={{2}^{3}}\times {{3}^{3}} \\
& 54={{2}^{1}}\times {{127}^{1}} \\
\end{align}\]
Now, for GCD,select each prime factor with the smaller exponent that is common to each of the prime factorization:
\[\begin{align}
& \text{GCD}={{2}^{1}} \\
& =2
\end{align}\]
Now, for LCM, select every prime factor that occurs, raised to the greater exponent in these prime factorizations:
\[\begin{align}
& \text{LCM}={{2}^{3}}\times {{3}^{3}}\times {{127}^{1}} \\
& =216\times 127 \\
& =27,432
\end{align}\]
Hence, the GCD is \[2\] and LCM is\[27,432\].
