Question

Find the LCM and HCF of $12$, $72$, $120$ using the prime factorization. Also, show that HCF$ \times $ LCM $ \ne $ product of the given three numbers.

Answer 1

Hint: There are various methods for finding the least common multiple and highest common factor of two given numbers. The simplest method to find LCM and HCF is by prime factorization method. Least common multiple is a product of common factors with highest power and all other non-common factors. HCF is the product of the lowest powers of all the common factors. We first break down the given numbers into their prime factors and then find their LCM and HCF according to the definitions.

Complete step-by-step solution:
The numbers given to us in the question are: $12$, $72$, $120$.
We have to find the least common multiple and highest common factor the given three numbers by prime factorization method.
In prime factorization, we represent the numbers as a product of their constituent prime factors. So, we get,
Prime factors of \[12\]$ = 2 \times 2 \times 3$
$ = {2^2} \times 3$
Prime factors of \[72\]$ = 2 \times 2 \times 2 \times 3 \times 3$
$ = {2^3} \times {3^2}$
Prime factors of \[120\]$ = 2 \times 2 \times 2 \times 3 \times 5$
$ = {2^3} \times 3 \times 5$
Now, Least common multiple is a product of common factors with highest power and all other non-common factors. So, finding the least common multiple of the three numbers, we get,
Least common multiple of $12$, $72$ and $120$$ = {2^3} \times {3^2} \times 5$
$ = 8 \times 9 \times 5$
$ = 360$
Hence, the least common multiple of $12$, $72$ and $120$ is $360$.
HCF is the product of the lowest powers of all the common factors.
Now, finding the Highest common factor of $12$, $72$ and $120$, we get,
Highest common factor of $12$, $72$ and $120$$ = {2^2} \times 3$
$ = 4 \times 3 = 12$
So, the highest common factor of $12$, $72$ and $120$ is $12$.
 So, the product of HCF and LCM $ = 12 \times 360 = 4320$
Now, the product of the three numbers $ = 12 \times 72 \times 120 = 103680$
So, we can clearly see that the product of LCM and HCF of the three numbers and the product of the three numbers is not equal. Hence, proved.

Note: Least common multiple (LCM) has wide ranging applications in real world as well as in mathematical questions. Knowledge of least common multiple is also used in addition and subtraction of fractions. Highest common factor is the greatest number that divides both the given numbers. Highest common factor can also be calculated by division method as well as by using Euclid’s division lemma.