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**Hint:**HCF means Highest Common Factor is the factor of any two or more numbers which are common among them. HCF of two numbers can be obtained by the Prime factorization method easily. In Prime factorization methods, factors of a number can be obtained by evenly dividing the number by a series of prime factors (2,3,5,7,11,13,17,19,23,29 .... ) until we obtain the quotient of 1 and remainder of 0. So, find all factors of both numbers. After that take out all common factors of both numbers. This is the HCF of the two numbers.

**Complete step-by-step solution:**

HCF means Highest Common Factor is the factor of any two or more numbers which are common among them. It is also called Greatest common factor (GCF) or Greatest common divisor (GCD).

HCF of two numbers can be obtained by the Prime factorization method easily. In Prime factorization methods factors of a number can be obtained by evenly dividing the number by a series of prime factors (2,3,5,7,11,13,17,19,23,29 .... ) until we obtain the quotient of 1 and remainder of 0.

So, firstly finding the factors of both numbers.

First number is 75.

Let's start dividing the number 75 by first prime number 2.

\[\dfrac{{75}}{2} = 37.5\]. Here the remainder is 1. So 75 is not evenly divided by 2.

Now trying with next prime number 3.

\[\dfrac{{75}}{3} = 25\]. Here the remainder is 0. So 3 is the first factor of 75.

Now, $\dfrac{{25}}{3} = 8.33$. Here the remainder is 1. So 25 is not evenly divided by 3.

Now trying with next prime number 5.

$\dfrac{{25}}{5} = 5$. Here the remainder is 0. So 5 is the second factor of 75.

Now, $\dfrac{5}{5} = 1$. Here remainder is 0 and quotient is 1. So 5 is the third factor of 75.

So number 75 can be represented by multiplying all prime factors of numbers.

So, $75 = 3 \times 5 \times 5$.

Now, Second number is 180.

Following the same step by step procedure as followed above for finding the factors of number 180.

Let's start dividing the number 180 by first prime number 2.

\[\dfrac{{180}}{2} = 90\]. Here the remainder is 0. So 2 is the first factor of 180.

Now, \[\dfrac{{90}}{2} = 45\]. Here the remainder is 0. So 2 is the second factor of 180.

Now, \[\dfrac{{45}}{2} = 22.5\]. Here the remainder is 1. So 45 is not evenly divided by 2.

Now trying with next prime number 3.

\[\dfrac{{45}}{3} = 15\]. Here the remainder is 0. So 3 is the third factor of 180.

Now, \[\dfrac{{15}}{3} = 5\]. Here the remainder is 0. So 3 is the fourth factor of 180.

Now, \[\dfrac{5}{3} = 1.67\]. Here the remainder is 2. So 5 is not evenly divided by 3.

Now trying with next prime number 5.

$\dfrac{5}{5} = 1$. Here remainder is 0 and quotient is 1. So 5 is the fifth factor of 180.

So, $180 = 2 \times 2 \times 3 \times 3 \times 5$.

Here in number 75 and 180 common is $3 \times 5$. SO HCF of the numbers 75 and 180 is $3 \times 5 = 15$.

**So, Option (D) is the correct answer.**

**Note:**HCF of two numbers can also be obtained by Division method. In this method first divide the large number by the small number. If the remainder is left, then divide the first divisor by remainder. If the remainder divides the first divisor completely, then the remainder is the HCF of given numbers. If the remainder does not divide the first divisor completely, then repeat the steps.

Ex,. Here given numbers are 75 and 180.

So, firstly dividing large numbers by small numbers, so $\dfrac{{180}}{{75}} = 2.4$. Here the remainder is 30.

Now, divide the first divisor 75 by first remainder 30, so $\dfrac{{75}}{{30}} = 2.5$. Here the remainder is 15.

So, divide the second divisor 30 by second remainder 15, so $\dfrac{{30}}{{15}} = 2$. Here the remainder is 0.

So, 15 is the HCF of the given numbers 75 and 180.

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