Answer
3456;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 48 and 72, begin with their prime factorization.
The factor tree indicates that
\[\begin{align}
& 48={{2}^{4}}\times 3 \\
& 72={{2}^{3}}\times {{3}^{2}}
\end{align}\]
First, clearly \[{{2}^{3}}\] can be taken out as common from \[{{2}^{3}}\]and \[{{2}^{4}}\].
Second, 3 can be taken out as common from 3and\[{{3}^{2}}\].
So, the greatest common divisor is provided by
\[{{2}^{3}}\times 3=24\]
Now, find the least common multiple of 48 and 72.
The prime factorization of 48 and 72 are as follows:
\[\begin{align}
& 48={{2}^{4}}\times 3 \\
& 72={{2}^{3}}\times {{3}^{2}}
\end{align}\]
The prime factors that occur are \[2\text{ and }3\].
The greatest exponent of \[2=4\].
Select \[{{2}^{4}}\].
The greatest exponent of \[3=2\].
Select \[{{3}^{2}}\].
The least common multiple is \[{{2}^{4}}\times {{3}^{2}}\].
\[\begin{align}
& {{2}^{4}}\times {{3}^{2}}=16\times 9 \\
& =144
\end{align}\]
Now, the product of the greatest common divisor and the least common multiple of 48 and 72 is \[24\times 144=3456\].
The product of 48 and 72 is \[48\times 72=3456\].
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.
