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Find the L.C.M and H.C.F of the following numbers. $0.48,\text{ }6.4,\text{ }{{0.256}^{{}}}$
(a)L.C.M $=19.2$ and H.C.F $=0.032$
(b)L.C.M $=6.4$ and H.C.F $=0.48$
(c)L.C.M $=20.2$ and H.C.F $=1.5$
(d)L.C.M $=12.9$ and H.C.F $=0.25$

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Hint: Convert each of the decimals to like decimals. In this question, we have to have three decimal places for each number to make them like decimals. Then, we can remove the decimal points. Find the H.C.F and L.C.M by prime factorization. And in the answer put the decimal points as there are a number of decimal places in the like decimals.

Complete step-by-step answer:
In this question we have been asked to find the L.C.M and H.C.F of the following numbers. $0.48,\text{ }6.4,\text{ }{{0.256}^{{}}}$.
We can notice that $0.48$ has two places of decimals, $6.4$ has one place of decimal and $0.256$ has three places of decimals.
So we have to convert all the three given numbers by adding three decimal places to make them like decimals.
Converting each of the decimals to like decimals we get $0.480,6.400,0.256$.
If we remove the decimal from all the numbers as $480,6400,256$.
We can write $480$ as the product of prime factors as $480=2\times 2\times 2\times 2\times 2\times 3\times 5$.
The product of prime factors of $6400$ is $6400=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 5\times 5$.
And the product of prime factors of $256$ is $256=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$.
We know that H.C.F is the greatest factor present in between the numbers. We can see that the prime factor 2 is repeating 5 times and is common to all numbers.
So we get H.C.F of $480,6400,256$ $=2\times 2\times 2\times 2\times 2=32$.
Putting the decimal points at three decimal places we get the H.C.F of $0.480,6.400,0.256$ is $0.032$.
L.C.M of the numbers is the least positive integer that is divisible by all those numbers. We can see that the prime factor 2 has the highest power of 8, 3 has the highest power of 1 and 5 has the highest power of 2. Now, we can multiply them together.
So we get L.C.M of $480,6400,256$ $=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\times 5\times 5=19200$.
Putting the decimal points at three decimal places the L.C.M of $0.480,6.400,0.256$ is $19.200=19.2$.
Hence the correct answer is option a.

Note: In this problem students must take care of changing the decimal point while finding the L.C.M and H.C.F of the decimal number from that of integers. Also students can use the long division method to find H.C.F and division method to find L.C.M. But since, it is difficult to compute using these methods, it is advised to use the prime factorisation method.
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