1.4Highest Common Factor (HCF)

The highest common factor (HCF) of two or more numbers is the largest number dividing exactly into all of them. Write each as a product of primes, then multiply the primes appearing in every factorisation, each at its smallest index. The smaller index is all the numbers can share.

HCF rule

\displaystyle \text{HCF} = \text{product of } \textbf{common} \text{ prime factors, each with the } \textbf{smallest} \text{ index}

One column per prime, one mark per index. The HCF takes each common prime at its smaller index.

Finding the HCF of two numbers

Find the HCF of $\displaystyle 84$ and $\displaystyle 126$ .

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\displaystyle 84

\displaystyle = 2^2 \times 3 \times 7

\displaystyle 126

\displaystyle = 2 \times 3^2 \times 7

Common prime factors:

2

3

and

7

Take the smaller index for each:

2^1

3^1

and

7^1

\displaystyle \text{HCF}

\displaystyle = 2 \times 3 \times 7

\displaystyle = 42

Now you try

Find the HCF of $200$ and $225$ .

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\displaystyle 200

\displaystyle = 2^{3} \times 5^{2} \quad \scriptsize\textit{prime factorisation}

\displaystyle 225

\displaystyle = 3^{2} \times 5^{2}

Common prime factors with lowest indices:

5^{2}

\displaystyle \text{HCF}

\displaystyle = 5^{2}

\displaystyle = 25

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Finding the HCF of three numbers

Find the HCF of $\displaystyle 72$ , $\displaystyle 108$ and $\displaystyle 180$ .

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\displaystyle 72

\displaystyle = 2^3 \times 3^2

\displaystyle 108

\displaystyle = 2^2 \times 3^3

\displaystyle 180

\displaystyle = 2^2 \times 3^2 \times 5

Common prime factors:

2

and

3

(the prime

5

appears only in

180

, so it is excluded).

Take the smallest index for each:

2^2

and

3^2

\displaystyle \text{HCF}

\displaystyle = 2^2 \times 3^2

\displaystyle = 36