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**Hint:**In this question, we find the greatest common factor of two numbers. We find the greatest common factor in the two methods. The first method is, we find the factor of given numbers. Then we choose the common numbers from the factorization of these numbers, the greatest common number is the greatest common factor of given two numbers. And the second method is, we find the prime factor of each number. Then choose the common numbers from prime factorization of given numbers. After that multiply, these numbers and the number which comes after multiplication is the greatest common factor.

**Complete step by step solution:**

In this question, the word greatest common factor is used. First, we know about the greatest common factor. The greatest common factor is defined as the largest positive integer which is divided by the given number without the reminder.

It is denoted as GCF.

Now we come to the question. To find the greatest common factor of two numbers, there are two methods.

First method:

First, we find the factor of given numbers \[20\] and \[30\] as below.

Now,

We find the factor of \[20\].

The factors of \[20\] are as below.

\[20 = 20,\;10,\;5,\;4,\;2,1.................\left( 1 \right)\]

Then,

We find the factor of \[30\].

The factors of \[30\] are as below.

\[30 = 30,\;15,\;10,\;6,\;5,\;3,\;2,\;1.................\left( 2 \right)\]

Now,

We find the common form equation \[\left( 1 \right)\;{\text{and}}\left( 2 \right)\].

Thus, the common numbers from the factors of \[20\] and \[30\] are as below.

\[ = 10,\;5,\;2,\;1\]

The greatest common number is \[10\].

Therefore, the greatest common factor of \[20\] and \[30\] is\[10\].

Second method:

We write the prime factorization of \[20\] and \[30\].

The prime factors of \[20\] are.

\[ \Rightarrow 20 = 2 \times 2 \times 5\]

The prime factors of \[30\] are.

\[ \Rightarrow 30 = 2 \times 3 \times 5\]

The common form factors are \[2\] and \[5\].

Then we multiply these two numbers.

\[ \Rightarrow 2 \times 5 = 10\]

**Therefore, the greatest common factor of \[20\] and \[30\] is \[10\].**

**Note:**

If you have two numbers and you want to find the greatest common factor of these two numbers. Then first you find the factor of the given two numbers. After that, you choose the common numbers from the factorization of given numbers. Thus, the greatest number from the common numbers of factorization is the greatest common factor.

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