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The LCM of two numbers is 1200. Which of the following cannot be their HCF?
A. 600
B. 500
C. 400
D. 200

Last updated date: 23rd Jun 2024
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Hint: First we will define both the terms that are HCF and LCM with examples. After that we will draw a relation between both the terms by analyzing their properties and definition that HCF is a factor of LCM, then we will use the given data in the question to get the answer.

Complete step by step answer:
Let’s first understand the meaning of HCF and LCM. The full forms of H.C.F. and L.C.M. are, Highest Common factor and Least Common Multiple, respectively. The H.C.F. defines the greatest factor present in between given two or more numbers for example let’s take two numbers that is 10 and 15, The H.C.F. of 10 and 15 will be 5 because the highest number that can divide both 10 and 15 is 5. Now, the L.C.M. defines the least number which is exactly divisible by two or more numbers, for example let us take 6 and 10.
Now, multiples of 6 are: 6,12,18,24,30.... and multiples of 10 are: 10,20,30,40.... We see that the common multiples for 10 and 6 are 30,60,90… and so on. The least common multiple in that lot would be 30. Hence, the LCM of 6 and 10 would be 30.
Now, we see that the LCM of two numbers will be always divisible by HCF of those two numbers because HCF will always be a factor of the given two numbers. For example let’s again take the numbers we took above that is 6 and 10, now here the LCM we found out is 30 and the highest common factor that can divide both the numbers will be 2. Now, we see that LCM that is 30 will be divisible by the HCF that is 2.
We will apply this same property onto our question. We have been given the LCM of two numbers now the HCF of two numbers will be a factor of the given LCM.
We have: $\text{LCM}=1200$ , now we will consider the options and see that 600,400,200 can be the HCF of the given two numbers as they are the factors of the LCM that means they divide it. Therefore, 500 cannot be the HCF of the given numbers.

So, the correct answer is “Option B”.

Note: Students can make mistakes when they use the relation between HCF and LCM that the product of any two given numbers say x, y will be equal to product of their HCF and LCM. That is $xy=HCF\times LCM$ , here one cannot draw a final conclusion and no proper proof is there to support the statement given.