In this question, we have to find the pair of numbers for which the LCM is 16.
LCM stands for Least Common Multiple, defined as the smallest multiple common number between the required numbers for which LCM is to be determined. It is the smallest positive number that is divisible by all given numbers. LCM can be determined between 2 or more than 2 numbers.
LCM can be found by three methods:
LCM by using prime factorization
LCM by using repeated division
LCM by using multiple
Here, we will find the LCM by using the method of multiples i.e. finding the common multiplies between the 2 given numbers and then selecting the smallest among them as the LCM for that pair.
Expert Answer
The LCM for each pair is calculated as follows
The LCM of $3$ and $16$ will be:
\[3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, …\]
\[16 = 16, 32, 48, …\]
Common Multiple is $48$. As it is the smallest common multiple, hence:
\[LCM = 48\]
The LCM of $2$ and $4$ will be:
\[2 = 2, 4, 6, 12, …\]
\[4 = 4, 8, 12, …\]
Common Multiples are $4,8, …$. As the smallest common multiple is $4$, hence
\[LCM = 4\]
The LCM of $4$ and $8$ will be:
\[4 = 4, 8, 12, 16, 20, 24, …\]
\[8 = 8, 16, 24, …\]
Common Multiples are $8,16, …$. As the smallest common multiple is $8$, hence
\[LCM = 8\]
The LCM of $4$ and $16$ will be:
\[4 = 4, 8, 12, 16, 20, 24, 28, 32, …\]
\[16 = 16, 32, …\]
Common Multiples are $16, 32, …$. As the smallest common multiple is $16$, hence
\[LCM = 16\]
Numerical Results:
So the required pair of numbers for which the LCM is $16$ is $4$ and $16$
Example:
Find out which of the following pairs has the LCM of $24$.
$a)$ $3$ and $8$
$b)$ $2$ and $12$
$c)$ $6$ and $4$$d)$ $4$ and $12$
Solution:
The LCM of $3$ and $8$ will be:
\[3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, …\]
\[8 = 8, 16, 24, 32, 40, 48, …\]
\[LCM = 24\]
The LCM of $2$ and $12$ will be:
\[2 = 2 ,4, 6, …\]
\[12 = 12, 24, 36, 48, 60, 72, …\]
\[LCM = 12\]
The LCM of $4$ and $6$ will be:
\[4 = 4, 8, 12, 16, 20, …\]
\[6 = 6, 12, 18, 24, …\]
\[LCM = 12\]
The LCM of $4$ and $12$ will be:
\[4 = 4, 8, 12, 16, 20, …\]
\[12 = 12, 24, 36, 48, 60, 72, …\]
\[LCM = 12\]
So the required pair is $3$ and $8$.
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