Question

Find the least common multiple of \[\dfrac{2}{7}\], \[\dfrac{3}{{14}}\] and \[\dfrac{5}{3}\].

Answer 1

Hint: There are various methods for finding least common multiple of the given numbers. The simplest method to find the least common multiple is by prime factorization method. In the prime factorization method, we first represent the given numbers as a product of their prime factors and then find the product of all the factors counting the common factors only once.

Complete step-by-step answer:
In the given question, we have to find the LCM of \[\dfrac{2}{7}\], \[\dfrac{3}{{14}}\] and \[\dfrac{5}{3}\].
The denominators of the given fractions are $ 7 $ , $ 14 $ and $ 3 $ .
To find the least common multiple of $ 7 $ , $ 14 $ and $ 3 $ , first we find out the prime factors of all the numbers.
Prime factors of \[7\] $ = 7 \times 1 $
Prime factors of \[14\] $ = 2 \times 7 $
Prime factors of \[3\] $ = 3 \times 1 $
Now, Least common multiple is a product of common factors with highest power and all other non-common factors. We can see that \[7\] is the only repeated factor in the given numbers.
Hence, least common multiple of $ 7 $ , $ 14 $ and $ 3 $ $ = 2 \times 3 \times 7 $
 $ = 42 $
Hence, the least common multiple of $ 7 $ , $ 14 $ and $ 3 $ is $ 42 $ .
So, the correct answer is “42”.

Note: Least common multiple (LCM) has wide ranging applications in real world as well as in mathematical questions. Knowledge of the least common multiple is also used in addition and subtraction of fractions. Highest common divisor is just a product of common factors with lowest power. Highest common factor can also be calculated by division method as well as by using Euclid’s division lemma.