Question

What is the Lowest Common Multiple (LCM) of $\dfrac{1}{5}$, $\dfrac{5}{8}$, $\dfrac{1}{3}$ and $\dfrac{3}{{16}}$ ?
A.$15$
B.$\dfrac{1}{{48}}$
C.$\dfrac{5}{4}$
D.$\dfrac{1}{4}$

Answer 1

Hint: Least Common Multiple or LCM is the smallest positive number that is a common multiple of two or more numbers. L.C.M of fraction numbers is equal to L.C.M of numerators divided by H.C.F of denominators. H.C.F is the highest common factor or we can say that H.C.F is the greatest number which divides each of the two or more numbers.
Formula: \[L.C.M{\text{ }} = \dfrac{{L.C.M{\text{ }}of{\text{ }}numerators}}{{H.C.F{\text{ }}of{\text{ }}denominators}}\]

Complete step-by-step answer:
Step 1: Find LCM of numerator: by calculating multiples and finding the lowest common multiple.
Now L.C.M of $1$, $1$, $3$, $5$:
The multiples of $1$ are: $1,2,3,4,5,6,.....$
The multiples of $3$ are: $3,6,9,12,15,18...$
The multiples of $5$ are: $5,10,15,20,25....$
$15$ is the lowest common multiple as it is a multiple common to all four.
Therefore, L.C.M of $\left( {1,3,5} \right) = 15$
Step 2: Find HCF of denominator.
Now H.C.F of $3$ , $5$ , $8$ ,$16$ :
The factors of $3$ are: $1,3$
The factors of $5$ are: $1,5$
The factors of $8$ are: $1,2,4,8$
The factors of $16$ are: $1,2,4,8,16$
$1$ is the only common factor as it is the only number which is common to all four.
Step 3: Put the values in the given formula.
\[L.C.M{\text{ }} = \dfrac{{L.C.M{\text{ }}of{\text{ }}numerators}}{{H.C.F{\text{ }}of{\text{ }}denominators}}\]
$ \Rightarrow L.C.M = \dfrac{{15}}{1}$
Or
$ \Rightarrow L.C.M = 15$
Thus the correct option is A.
Therefore, Lowest Common Multiple (LCM) of $\dfrac{1}{5}$ , $\dfrac{5}{8}$ , $\dfrac{1}{3}$ and $\dfrac{3}{{16}}$ is $15$.
So, the correct answer is “15”.

Note: We can find the H.C.F of any given numbers by using two methods: by prime factorization method and by division method. There are various methods to find L.C.M also, such as prime factorization, ladder method, calculating multiples, and finding the common multiple. Whatever method we follow we will get the same answer.