Question

Find HCF of \[30\] and \[210\] by using the prime factorization method.

Answer 1

Hint: In this particular problem we have to find the highest common factor. So first we find all the factors of a given number and after that choose the common factor in both numbers and the product of all common factors gives the highest common factor to us.

Complete step-by-step answer:
Follow the below-given steps to find the highest common factor of numbers using prime factorization method.
Step 1: Write each number as a product of its prime factors. This method is called here prime factorization.
Step 2: Now list the common factors of both the numbers
Step 3: The product of all common prime factors is the Highest common factor (use the lower power of each common factor)
So, here we go with first step:
Factor of \[{\mathbf{30}}\] :
\[30 = 2 \times 3 \times 5\]
Now similarly we find factor of \[210\] :
$210 = 2 \times 3 \times 5 \times 7$
So now we go on to the next step where we have to collect the common factor of both the numbers.
So in $2 \times 3 \times 5$ and $2 \times 3 \times 5 \times 7$
Common factor is \[2,3,5\] .
Now in the next step we have to make a product of the entire common factor that is our required answer.
So product of $2 \times 3 \times 5$ gives Highest common factor
\[HCF{\text{ }}\left( {30,210} \right){\text{ }} = {\text{ }}30.\]

The HCF of 30 and 210 is 30.

Note: We can solve this problem by using a long division method where \[210\] is completely divided by \[30\] Hence, it is the highest common factor, If we go with the prime factorization method we can collect only common factors. Hence, we can say that the key to determine the highest common factor is to determine the factorization and if we are having the same number then the highest common factor will be the same because they are having the same factorization.