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how to find hcf and LCM of a given number .explain with example

The above given Explanation is correct.

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The HCF and LCM of any given set of numbers can be find out by representing the given numbers in terms of their prime factors.

Example - 

the numbers 48, 54 and 45 are represented in terms of prime factors as:

48 = 2 × 2 × 2 × 2 × 3 

54 = 2 × 3 × 3 × 3

45 = 3 × 3 × 5

Now, HCF is the product of all common numbers in the given set of numbers..

Examlpe -  

the only common number in 48, 54 and 45 is 3.

Therefore,  HCF of 48, 54 and 45 = 3 

Similarly, LCM is the product of all common and uncommon numbers in the given set of numbers.

Therefore,  LCM of 48, 54 and 45 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 = 2160

Similarly, LCM and HCF of 123 and 567 can be find out by finding the product of prime factors as:

123 = 3 × 41

and 567 = 3 × 3 × 3 × 3 × 7  

So, HCF = 3

and LCM = 3 × 3 × 3 × 3 × 7 × 41 = 23247

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 The HCF of two or more numbers is the largest common factor of the given numbers.

The LCM is the smallest number that is a common multiple of two or more numbers.
 
Let us take an example:
Anurag takes 6 minutes to complete one round of jogging around the circular track of a park, while Twinkle takes 8 minutes to do the same. If both of them start jogging at the same time from the same point, then how much time will they take before they meet at the point from which they started?
 
Its solution can be given as,
Anurag and Twinkle take 6 minutes and 8 minutes respectively to complete one round of the circular track.
Anurag jogs around the circles after every 6 minutes i.e., he completes the one, two, three, four rounds around the park after 6 minutes, 12 minutes, 18 minutes, 24 minutes, 30 minutes respectively and so on.
Twinkle jogs around the circles after every 8 minutes i.e., she completes the one, two, three, four rounds around the park after 8 minutes, 16 minutes, 24 minutes, 32 minutes, 40 minutes respectively and so on.
You may observe that the smallest time after which they will meet at the starting point is 24 minutes. And you also know that 24 is the L.C.M. of 6 and 8.
Now, the thing is that without analysing what I have done above, how will you know that this question is based on L.C.M.?
It is quite simple. The time after which they meet must be greater than the time taken by them to complete one round. During that time, they have completed a number of rounds.
From this it is clear that the required time must be a common multiple of 6 minutes and 8 minutes. Since, you are required to find the least time, so, the required time must a least common multiple of 6 minutes and 8 minutes, I,e, L.C.M. of 6 minutes and 8 minutes.
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