Question

Find the LCM and HCF of the following pairs of integers and verify that LCM x HCF = Product of two numbers.
(i) 336 and 54

Answer 1

Hint – In order to solve this problem you need to know that HCF is the highest common factor among the given numbers and LCM is the lowest common factor among the given numbers. Knowing this will solve your problem and will give you the right answer.

Complete step-by-step solution -
LCM - In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b.
HCF - The highest common factor is found by multiplying all the factors which appear in both lists: So the HCF of 60 and 72 is $2 \times 2 \times 3$ which is 12. The lowest common multiple is found by multiplying all the factors which appear in either list: So the LCM of 60 and 72 is $2 \times 2 \times 2 \times 3 \times 3 \times 5$ which is 360.
We have to find the LCM and HCF of 336 and 54.
So first, calculate the factors of each number given
So, factors of $336 = 2 \times 2 \times 2 \times 2 \times 7 \times3 \times1……………(1)$
Now the factors of $54 = 2 \times 3 \times 3 \times 3……………….(2)$
So, HCF means the highest common factors, from equation 1 and 2 highest common factors of 336 and 54 are 2 and 3 thus $HCF = 2 \times 3=6$
So, LCM means the lowest common factor, from equations 1 and 2 the lowest common factors of 336 and 54 are 2 (4 times), 3( 3 times), and 7 (one time).
HCF = ${2^4} \times {3^3} \times {7^1} = 3024.$
Now we need to prove that LCM x HCF is the product of two numbers.
So, on putting the values we get, $6 \times 3024=18114$ and the product of those two numbers is
$336 \times 54 = 18114.$
Hence, it is proved that LCM x HCF = Product of two numbers.

Note – All questions of finding the LCM and HCF involve the same concept of finding the factors of the numbers so that their highest common factors and lowest common factors can be calculated and then solved as per the questions guidelines to reach the solution. Doing this will solve your problem and will give you the right answer.