Finding the Highest Common Factor and Lowest Common Multiple

Welcome back. In the previous lesson, you expressed positive integers as products of prime factors, keeping repeated primes through indices such as

$$1260=2^2\times 3^2\times 5\times 7.$$

That factorisation is now a tool rather than an endpoint. This lesson uses it to find the highest common factor (HCF) and lowest common multiple (LCM) of two or more integers. These ideas will later support simplifying fractions, working with algebraic factors, and solving problems involving repeating cycles or equal-sized groups.

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Factors, multiples, and the two questions to distinguish

A factor divides a number exactly. For example, $6$ is a factor of $18$, because

$$18÷6=3.$$

A multiple is obtained by multiplying a number by a whole number. For example, $18,36,$ and $54$ are multiples of $18$.

The difference matters:

If the problem asks…	Think about…	Useful result
“What is the largest size that divides every quantity exactly?”	factors	HCF
“What is the first time or smallest amount that works for every quantity?”	multiples	LCM

The highest common factor of two or more positive integers is the greatest positive integer that divides every number exactly.

For example, the factors of $12$ and $18$ are:

$$\begin{array}{rl}12& :1,2,3,4,6,12,\\ 18& :1,2,3,6,9,18.\end{array}$$

Their common factors are $1,2,3,$ and $6$. Therefore,

$$\mathrm{HCF}(12,18)=6.$$

The lowest common multiple is the smallest positive integer that is a multiple of every given number. The multiples of $10$ and $12$ begin as:

$$\begin{array}{rl}10& :10,20,30,40,50,60,\dots \\ 12& :12,24,36,48,60,\dots \end{array}$$

The first positive multiple they share is $60$, so

$$\mathrm{LCM}(10,12)=60.$$

For small numbers, listing factors or multiples is a sensible way to see the ideas. For larger numbers, however, prime factorisation is more reliable and much faster.

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Seeing prime factors as shared and unshared “ingredients”

Suppose we want both the HCF and LCM of $18$ and $30$.

First prime-factorise:

$$18=2\times 3\times 3$$

and

$$30=2\times 3\times 5.$$

The numbers share one $2$ and one $3$. The extra $3$ belongs only to $18$; the $5$ belongs only to $30$.

The diagram provides two complementary readings:

• To find the HCF, multiply only the factors in the overlap:

$$\mathrm{HCF}(18,30)=2\times 3=6.$$

• To find the LCM, multiply every prime factor appearing anywhere in the diagram, but only once per displayed copy:

$$\mathrm{LCM}(18,30)=2\times 3\times 3\times 5=90.$$

Check the meanings:

$$18÷6=3,30÷6=5,$$

so $6$ divides both numbers. Also,

$$90÷18=5,90÷30=3,$$

so $90$ is a multiple of both numbers.

The Venn diagram is not just a memory trick. It represents a central fact:

• The HCF contains only prime factors that every number can supply.
• The LCM contains every prime factor that at least one number requires.

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The prime-factorisation method: minimum exponents and maximum exponents

A Venn diagram is useful for two numbers, but an exponent table is clearer once factors repeat or three or more integers are involved.

Consider $84$ and $280$.

$$84=2^2\times 3\times 7$$ $$280=2^3\times 5\times 7.$$

Write every prime that appears in either factorisation. A missing prime has exponent $0$.

Prime	Exponent in $84$	Exponent in $280$	HCF exponent	LCM exponent
$2$	$2$	$3$	$2$	$3$
$3$	$1$	$0$	not included	$1$
$5$	$0$	$1$	not included	$1$
$7$	$1$	$1$	$1$	$1$

For the HCF, take the smallest exponent for each prime, but only if that exponent is positive:

$$\mathrm{HCF}(84,280)=2^2\times 7=28.$$

For the LCM, take the largest exponent for every prime that appears:

$$\mathrm{LCM}(84,280)=2^3\times 3\times 5\times 7=840.$$

Why minimum for HCF?

A common factor must divide both numbers. It cannot contain more copies of a prime than the number with fewer copies.

For instance, $84$ contains $2^2$, not $2^3$. Therefore a common factor cannot contain $2^3$, even though $280$ does. The most both numbers can supply is $2^2$.

Why maximum for LCM?

A common multiple must be divisible by both numbers. Since $280$ contains $2^3$, the LCM must contain at least $2^3$. Taking fewer copies would prevent divisibility by $280$. Taking more copies would create a larger multiple than necessary.

So the two rules are exact opposites:

$$\boxed{\text{HCF: common primes with smallest exponents}}$$ $$\boxed{\text{LCM: all primes with largest exponents}}$$

A dependable written method is:

1. Prime-factorise every integer.
2. List each prime appearing in at least one factorisation.
3. For the HCF, keep only primes appearing in every factorisation and select their smallest exponents.
4. For the LCM, keep every listed prime and select its largest exponent.
5. Multiply and check divisibility.

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Interpreting HCF and LCM in context

The arithmetic method is the same, but the wording of a problem tells you which result is needed.

An HCF problem usually involves equal division with no remainder, together with a request for the greatest possible size, length, or number of groups.

For example, a rectangular floor is $570\text{ cm}$ by $450\text{ cm}$. What is the largest square tile side length that covers the floor without cutting tiles?

$$570=2\times 3\times 5\times 19$$ $$450=2\times 3^2\times 5^2.$$

The common primes with the smallest exponents are $2$, $3$, and $5$:

$$\mathrm{HCF}(570,450)=2\times 3\times 5=30.$$

So the largest possible square tile has side length

$$30\text{ cm}.$$

An LCM problem usually involves repeating intervals, synchronisation, or the smallest shared total.

Suppose one event happens every $6$ days and another happens every $10$ days. If both happen today, when will they next happen together?

$$6=2\times 3,10=2\times 5.$$

Take every prime at its largest required exponent:

$$\mathrm{LCM}(6,10)=2\times 3\times 5=30.$$

They next coincide after

$$30\text{ days}.$$

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Extending the method to three integers

The prime-factor method scales directly to three or more integers.

Find the HCF and LCM of $24$, $42$, and $30$.

$$24=2^3\times 3$$ $$42=2\times 3\times 7$$ $$30=2\times 3\times 5.$$

Every number contains one factor $2$ and one factor $3$. No other prime occurs in all three factorisations. Therefore,

$$\mathrm{HCF}(24,42,30)=2\times 3=6.$$

For the LCM, include all primes needed by at least one number, using the largest exponent required:

$$\mathrm{LCM}(24,42,30)=2^3\times 3\times 5\times 7=840.$$

A fast check confirms the LCM:

$$840÷24=35,840÷42=20,840÷30=28.$$

Each quotient is a whole number, so $840$ is a common multiple. Because no prime exponent was made larger than necessary, it is the least common multiple.

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Checks and common errors

Before accepting an answer, use the definition to check it.

For an HCF:

• Does it divide every given integer exactly?
• Is there a larger common factor?
• If the HCF is $1$, are the numbers coprime? Coprime means their only positive common factor is $1$.

For an LCM:

• Is it divisible by every given integer?
• Did you include every prime needed by at least one number?
• Did you use the largest exponent, rather than multiplying repeated prime factors unnecessarily?

The most frequent errors are these:

Error	Why it fails	Correction
Using the largest exponent for the HCF	One number may not contain enough copies of that prime.	Use the smallest shared exponent.
Using the smallest exponent for the LCM	The result may fail to be divisible by one of the numbers.	Use the largest exponent needed.
Omitting a prime that appears in only one factorisation	The LCM then cannot be divisible by that number.	Include all primes for the LCM.
Multiplying the original integers to get the LCM	This produces a common multiple, but often not the least one.	Combine prime factors using maximum exponents.
Confusing group count with items per group	The HCF often gives the greatest number of equal groups; division then gives the contents of each group.	State what the HCF represents in context.

For two positive integers only, there is also a useful verification identity:

$$\mathrm{HCF}(a,b)\times \mathrm{LCM}(a,b)=a\times b.$$

For $84$ and $280$,

$$28\times 840=84\times 280.$$

This is a check, not the main method. It does not extend in the same simple form to three or more integers.

In your error log, record the reason for any incorrect result. For example, write “used $2^3$ in the HCF although one number contained only $2^2$” rather than simply “HCF error.” That diagnosis is what makes later review effective.

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Key takeaways

HCF and LCM are built from the same prime factorisations but answer opposite structural questions.

• The HCF is the largest number that divides every given integer exactly.
• The LCM is the smallest positive number divisible by every given integer.
• For small values, lists of factors or multiples can reveal the answer.
• For larger values, prime factorisation is the systematic method.
• For the HCF, use only shared primes and take the smallest exponent.
• For the LCM, use every prime that appears and take the largest exponent.
• Equal sharing, largest tiles, and longest exact cuts point to HCF.
• Repeating schedules, first coincidences, and minimum shared lengths point to LCM.

Next, you will use prime factorisation once more, this time to determine exact square roots and cube roots by grouping repeated prime factors.