Answer
648;the product of the greatest common divisor and the least common multiple of two numbers is equal to the product of those two numbers.
Work Step by Step
To find the greatest common divisor of 24 and 27, begin with their prime factorization.
The factor tree indicates that
\[\begin{align}
& 24={{2}^{3}}\times 3 \\
& 27={{3}^{3}}
\end{align}\]
Now, 3 is a common prime number to both factorizations.
The exponents of 3 are 1 and 3.
So, select \[{{3}^{1}}\]
The greatest common divisor \[{{3}^{1}}=3\].
Therefore, the greatest common divisor is 3.
Now, find the least common multiple of 24 and 27.
The prime factorization of 24 and 27 are as follows:
\[\begin{align}
& 24={{2}^{3}}\times 3 \\
& 27={{3}^{3}}
\end{align}\]
The prime factors that occur are \[2\text{ and }3.\]
The greatest exponent of \[3=3\].
Select \[{{3}^{3}}\].
The least common multiple is \[{{2}^{3}}\times {{3}^{3}}\].
\[\begin{align}
& {{2}^{3}}\times {{3}^{3}}=8\times 27 \\
& =216
\end{align}\]
Now, the product of the greatest common divisor and the least common multiple of 24 and 27 is \[3\times 216=648\].
The product of 24 and 27 is \[24\times 27=648\].
Therefore, the product of the greatest common divisor and the least common multiple of two numbers is equal to the product of those two numbers.
