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Which pair of numbers has an LCM of $16$

  • $3$ 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:

  1. LCM by using prime factorization
  2. LCM by using repeated division
  3. 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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