Factors, Prime Factors and Highest Common Factor (HCF)

 
2.1  Factors, Prime Factors and Highest Common Factor (HCF)
 
Factors
  • A number that divides another number completely.
  • The factors of \(12\) are \(1, 2,3,4,6\) and \(12\).
 
Example

Is \(12\) the factor of \(36\)?

\(36\div 12=3\)

Thus, \(12\) is the factor of \(36\).

 
Prime Factor

Factor that is a prime number.

 
Example

The factors of \(18\) are \(1, 2,3,6,9\) and \(18\).

Between these factors, \(2\) and \(3\) are prime numbers.

Thus, the prime factors of \(18\) are \(2\) and \(3\).

 
Solution Methods
 
Repeated division:
 
Express \(60\) in the form of prime factorisation.
 

Perform division repeatedly by dividing with the smallest prime number.

Division is continued until the quotient is \(1\).

\(\begin{array}{c} 2\\2\\3\\5 \\\phantom{-} \end{array} \begin{array}{|c} \quad60\quad\\ \hline \quad30\quad\\ \hline \quad15\quad\\ \hline \quad5\quad\\ \hline \quad1\quad\\ \end{array} \begin{array}{c}\end{array}\\\\\)

Thus, 

\(60=2\times2\times3\times5\).

 
Factor trees:
 
Express \(60\) in the form of prime factorisation.
 

 

Thus, 

\(60=2\times2\times3\times5\).

 
Common Factor

A number that is a factor of a few other numbers.

 
Example

Determine whether \(6\) is a common factor of \(24\) and \(36\).

\(24\div6=4 \\36\div6=6\)

\(24\) and \(36\) can be divided completely by \(6\).

Thus, \(6\) is a common factor of \(24\) and \(36\).

 
Highest Common Factor (HCF)

The greatest number among the common factors.

 
Example

(i) Determine the highest common factor of \(18\) and \( 24\).

Listing the common factors:

Factors of \(18 : 1 , 2 , 3 , 6 , 9, 18\)

Factors of \(24 : 1 , 2 , 3 , 4 , 6 , 8, 12, 24\)

So, the common factors of \(18 \) and \(24\) is \(1, 2, 3 \) and \(6\).

Thus, HCF is \(6\).

 

(ii) Determine the highest common factor of \(30,60\) and \( 72\).

Repeated division:

\(\begin{array}{c} 2\\2\\3 \\\phantom{-} \end{array} \begin{array}{|c} \quad36,\,60,\,72\quad\\ \hline \quad18,\,30,\,36\quad\\ \hline \quad9,\,15,\,18\quad\\ \hline \quad3,\,5,\,6\quad\\ \end{array} \begin{array}{c}\end{array}\\\\\)

Thus, HCF of \(36, 60\) and \(72\) is

\(2\times2\times3 = 12\).

 

(iii) Determine the highest common factor of \(48,64\) and \(80\).

Prime factorisation:

\(48 = 2\times2\times 2\times2\times3\)

\(64 = 2 × 2 × 2 × 2 × 2 × 2 \)

\(80 = 2\times2\times 2\times 2\times5\)

Thus, HCF is

\(2 × 2 × 2 × 2 = 16\).

Factors, Prime Factors and Highest Common Factor (HCF)

 
2.1  Factors, Prime Factors and Highest Common Factor (HCF)
 
Factors
  • A number that divides another number completely.
  • The factors of \(12\) are \(1, 2,3,4,6\) and \(12\).
 
Example

Is \(12\) the factor of \(36\)?

\(36\div 12=3\)

Thus, \(12\) is the factor of \(36\).

 
Prime Factor

Factor that is a prime number.

 
Example

The factors of \(18\) are \(1, 2,3,6,9\) and \(18\).

Between these factors, \(2\) and \(3\) are prime numbers.

Thus, the prime factors of \(18\) are \(2\) and \(3\).

 
Solution Methods
 
Repeated division:
 
Express \(60\) in the form of prime factorisation.
 

Perform division repeatedly by dividing with the smallest prime number.

Division is continued until the quotient is \(1\).

\(\begin{array}{c} 2\\2\\3\\5 \\\phantom{-} \end{array} \begin{array}{|c} \quad60\quad\\ \hline \quad30\quad\\ \hline \quad15\quad\\ \hline \quad5\quad\\ \hline \quad1\quad\\ \end{array} \begin{array}{c}\end{array}\\\\\)

Thus, 

\(60=2\times2\times3\times5\).

 
Factor trees:
 
Express \(60\) in the form of prime factorisation.
 

 

Thus, 

\(60=2\times2\times3\times5\).

 
Common Factor

A number that is a factor of a few other numbers.

 
Example

Determine whether \(6\) is a common factor of \(24\) and \(36\).

\(24\div6=4 \\36\div6=6\)

\(24\) and \(36\) can be divided completely by \(6\).

Thus, \(6\) is a common factor of \(24\) and \(36\).

 
Highest Common Factor (HCF)

The greatest number among the common factors.

 
Example

(i) Determine the highest common factor of \(18\) and \( 24\).

Listing the common factors:

Factors of \(18 : 1 , 2 , 3 , 6 , 9, 18\)

Factors of \(24 : 1 , 2 , 3 , 4 , 6 , 8, 12, 24\)

So, the common factors of \(18 \) and \(24\) is \(1, 2, 3 \) and \(6\).

Thus, HCF is \(6\).

 

(ii) Determine the highest common factor of \(30,60\) and \( 72\).

Repeated division:

\(\begin{array}{c} 2\\2\\3 \\\phantom{-} \end{array} \begin{array}{|c} \quad36,\,60,\,72\quad\\ \hline \quad18,\,30,\,36\quad\\ \hline \quad9,\,15,\,18\quad\\ \hline \quad3,\,5,\,6\quad\\ \end{array} \begin{array}{c}\end{array}\\\\\)

Thus, HCF of \(36, 60\) and \(72\) is

\(2\times2\times3 = 12\).

 

(iii) Determine the highest common factor of \(48,64\) and \(80\).

Prime factorisation:

\(48 = 2\times2\times 2\times2\times3\)

\(64 = 2 × 2 × 2 × 2 × 2 × 2 \)

\(80 = 2\times2\times 2\times 2\times5\)

Thus, HCF is

\(2 × 2 × 2 × 2 = 16\).