Question

The HCF of the numbers $0.48$, $0.72$ and $0.108$ is-
A) $1$
B) $12$
C) $0.12$
D) $0.012$

Answer 1

Hint:
First convert the given decimals into like decimals. Then remove the decimal point and find the HCF using the prime factorization method. Then put the decimal point the same as there are a number of decimal places in the given like decimals to get the answer.

Complete step by step solution:
First we will convert the given decimals into like decimals which will make it easier to find the HCF by adding zeros after the last number on the right of the decimal point. This does not change the value of numbers. So we can write the given decimals as-
$ \Rightarrow 0.480$, $0.720$ and $0.108$
Now on removing the decimal point we get- $480$, $720$ and $0.108$
Now, we will find the HCF of these numbers using prime factorization method as follows-
First we find the prime factors of each of the given numbers. We know the smallest prime number is $2$ so first we will divide the numbers by it and continue dividing the obtained number again by $2$ till it cannot be divided by it or we get $1$. If the number is then not divisible by $2$, try the next prime number which will divide it completely.
So the prime factors are written as-
Prime factors of $480 = \underline {2 \times 2} \times \underline {2 \times 2} \times 2 \times 3 \times 5$
Prime factors of $720 = \underline {2 \times 2} \times \underline {2 \times 2} \times \underline {3 \times 3} \times 5$
And Prime factors $108 = \underline {2 \times 2} \times \underline {3 \times 3} \times 3$
Now identify the common prime factors of the given numbers. The common prime factors of the given numbers are-$2 \times 2 \times 3$
Now multiply all the common factors to find the HCF of the given numbers.
HCF=$2 \times 2 \times 3 = 12$
Now on putting the decimal points before three digits of the HCF, we get-
HCF=$0.012$

The correct answer is option D.

Note:
Prime factorization method is very important in cryptography, which is the study of secret codes. Since these secret codes are based on numbers, so to break them we find prime factors of the large numbers as factoring large numbers is very hard and takes a computer a long time to do it. There are also other methods to find HCF. We can find HCF using long division method-
In this method, the highest number is multiplied by the smallest number.
Then the smallest number is divided by the remainder left. Again if we get remainder, then the divisor is divided by this remainder.
This process continues till there is no remainder left and the last quotient obtained is the HCF.