Question

Find $HCF$ and $LCM$ of $336$ and $54$ by prime factorisation and verify that?

Answer 1

Hint: Here in this question, we have to find the Highest common factor (HCF) and least common multiple (LCM) of given numbers by prime factorization method. Prime factorization of a number means to express the given number as a product of the prime factors, that is, the product of the numbers that are prime numbers and divide the given number completely. We can find out the correct answer using this information.

Complete step by step answer:
The Least Common Multiple (LCM) of two numbers is the smallest number that has both of the first two numbers as factors. The largest positive integer which divides two or more integers without any remainder is called Highest Common Factor (HCF) or Greatest Common Divisor or Greatest Common Factor (GCF).

Prime factorization is the process of finding the prime numbers, which are multiplied together to get the original number. A number that is not divisible by any other number except 1 and itself is known as a prime number. For example, 5 is not divisible by any number other than 1 and 5. Consider the given numbers 336 and 54.Let us find the prime factorisation of 336 & 54:
\[336 = 2 \times 2 \times 2 \times 2 \times 3 \times 7\]
\[\Rightarrow 336 = {2^4} \times 3 \times 7\]
And
\[54 = 2 \times 3 \times 3 \times 3\]
\[\Rightarrow 54 = 2 \times {3^3}\]

The highest common factor (HCF) of 336 and 54 is
HCF of \[\left( {336,54} \right) = 2 \times 3\]
\[\therefore \] HCF of \[\left( {336,54} \right) = 6\]
The Least Common Multiple (LCM) of 336 and 54 is
LCM of \[\left( {336,54} \right) = {2^4} \times {3^3} \times 7 \\ \]
LCM of \[\left( {336,54} \right) = 16 \times 27 \times 7 \\ \]
\[\therefore \] LCM of \[\left( {336,54} \right) = 3024 \\ \]
To verify, as we know, the product of HCF and LCM of given two numbers will be equal to the product of given two numbers.
\[HCF\left( {336,54} \right) \times LCM\left( {336,54} \right) = 336 \times 54\]
\[ \Rightarrow \,\,\,6 \times 3024 = 336 \times 54\]
\[\therefore \,\,\,18144 = 18144\]
Hence, verified.

Note: By factoring a number we mean to express it as a product of two numbers or it can also be defined as the division of a given number by some other number such that the remainder is zero. Usually, negative factors of a number are not considered and the fractions can never be considered as a factor of a number. Factors of a number are defined as all the possible sets of two numbers whose product is equal to the given number. Thus, we should remember the difference between a factor and prime factor and solve a question by keeping the mentioned information in mind.