$3$ and $16$
$2$ and $4$
$4$ and $8$
$4$ and $16$
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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