Question

Find the highest common factor of \[36\] and \[84\].
A. \[4\]
B. \[6\]
C. \[12\]
D. \[18\]

Answer 1

Hint: Here we will use the factorization method for finding the H.C.F(Highest common factor) of two numbers in which we will write the factors of all the numbers given from which the highest common number among them will be the HCF of those numbers. It is denoted as \[{\text{H}}{\text{.C}}{\text{.F}}\left( {a,b} \right)\], where \[a\] and \[b\] are two numbers. For example:
HCF of \[4\] and \[12\] are as given below:
Factors of
\[4 = 2,\]\[4\],\[1\]
Factors of
\[12 = 2,\]\[3\],\[4\], \[6\], \[12\]
Common factors \[ = 1,\]\[2\], \[4\]
HCF of \[4\] and \[12\]\[ = 4\]

Complete step-by-step solution:
Step 1: First of all, we will write the factors of both the numbers \[36\] and \[84\] as shown below:
Factors
\[36\] are \[1\], \[2\], \[3\], \[4\], \[6\], \[9\],\[12\], \[18\], \[36\]
Factors of \[84\] are \[1\], \[2\],\[3\], \[4\],\[6\], \[7\],\[12\], \[14\],\[21\],\[28\], \[42\], \[84\]
Step 2: We will select the common factors from the factors of
\[36\] and \[84\]as shown below:
\[ \Rightarrow {\text{Common factors}} = 1,2,3,4,6,12\]
The greatest common factor among the common terms is \[12\].
So, we can say that
\[{\text{H}}{\text{.C}}{\text{.F}}\left( {36,84} \right) = 12\]

Option C is the correct answer.

Note: Students need to remember that there are many ways for finding the HCF of any two or more numbers. For example:
Factorization Method: In which we used to find all possible factors of the numbers and after that, the highest common factor among them will be our HCF.
Prime Factorization Method: In this method, we will list out all the prime factors of each number, for example, we need to find the HCF of \[12\] and
\[36\]. So, the prime factors are:
The prime factorization of
\[12\]\[ = 2 \times 2 \times 3\]
The prime factorization of \[36\]\[ = 2 \times 2 \times 3 \times 3\]
The common prime factors of
\[12\] and \[36\] are \[2\], \[2\] and \[3\].
So, by multiplying the common prime factors, we can find the HCF as shown below:
The greatest common factor of \[12\] and \[36\] is \[2 \times 2 \times 3 = 12\].