
Find the LCM and HCF of , , using the prime factorization. Also, show that HCF LCM product of the given three numbers.
Answer
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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: , , .
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
Prime factors of
Prime factors of
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 , and
Hence, the least common multiple of , and is .
HCF is the product of the lowest powers of all the common factors.
Now, finding the Highest common factor of , and , we get,
Highest common factor of , and
So, the highest common factor of , and is .
So, the product of HCF and LCM
Now, the product of the three numbers
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.
Complete step-by-step solution:
The numbers given to us in the question are:
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
Prime factors of
Prime factors of
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
Hence, the least common multiple of
HCF is the product of the lowest powers of all the common factors.
Now, finding the Highest common factor of
Highest common factor of
So, the highest common factor of
So, the product of HCF and LCM
Now, the product of the three numbers
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.
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