Question

Find the least common multiple of \[10\] and \[12\].

Answer 1

Hint: There are various methods for finding the least common multiple of two 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 two 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:
To find the least common multiple of \[10\] and \[12\], first we find out the prime factors of both the numbers.
Prime factors of \[10\]$ = 2 \times 5$
Expressing this in exponential form, we get, $10={2^1} \times {5^1}$
Prime factors of \[12\]$ = 2 \times 2 \times 3$
Expressing in exponential form, we get, $12={2^2} \times {3^1}$
Now, Least common multiple is a product of common factors with highest power and all other non-common factors. We can see that $2$ is the only common factor of \[10\] and \[12\].
The least common multiple of \[10\] and \[12\]$ is equal to {2^2} \times 3 \times 5$
$=4 \times 3 \times 5$
$ = 60$

Therefore, the least common multiple of \[10\] and \[12\] is $60$.

Note: We know that the product of two numbers is equal to the product of the LCM and HCF of the two numbers. Alternative way to find LCM of the two numbers is to first find the HCF of the numbers and then substituting it in the formula:
\[HCF \times LCM = {\text{Product}}\,{\text{of}}\,{\text{numbers}}\]
Highest common divisor is just a product of common factors with lowest power.
We know that prime factors of \[10\]$ = 2 \times 5$
Prime factors of \[12\]$ = 2 \times 2 \times 3$
So, we get the HCF of the numbers \[10\] and \[12\] is $2$.

Substituting HCF of the numbers in the formula, we get,
\[ \Rightarrow 2 \times LCM = {\text{10}} \times {\text{12}}\]
Dividing both sides by two, we get,
\[ \Rightarrow LCM = \dfrac{{{\text{10}} \times {\text{12}}}}{2}\]
Simplifying the calculations, we get,
\[ \Rightarrow LCM = \dfrac{{{\text{120}}}}{2}\]
Cancelling common factors in numerator and denominator, we get,
\[ \Rightarrow LCM = 60\]
So, we get the LCM of \[10\] and \[12\] is \[60\].