You now know how to find HCF using common primes with lowest exponents.

LCM is the complement: ALL primes with HIGHEST exponents.

Notice the symmetry in the questions we ask:

Concept	The Core Question
HCF	What is the largest number that fits into both?
LCM	What is the smallest number that both fit into?

The prime factorisation method makes this symmetry visible.

In this section, you will:

Master LCM computation
Learn a powerful verification tool: the product property

1. Contrasting HCF and LCM Rules

You now know how to find HCF using common primes with lowest exponents.

LCM is the complement:

ALL primes
with HIGHEST exponents

Concept	The Key Question
HCF	What is the largest number that fits into both?
LCM	What is the smallest number that both fit into?

These two questions have symmetric answers, and the prime factorisation method makes this symmetry visible.

The most common error in this cluster is confusing the HCF and LCM rules. Before computing LCM, let's make sure you can clearly distinguish the two rules and explain why they differ.

You have learned two rules:

HCF rule: Product of common primes with lowest exponents.
LCM rule: Product of all primes with highest exponents.

✍️ Question

State the two key differences between the HCF and LCM rules.

For each difference, explain WHY the rules differ.

Hint: Think about what HCF and LCM must satisfy.

Let me explain the two rules side by side and WHY they differ.

To understand the logic, we have to look at what each one is trying to achieve. One looks for what is shared, and the other looks for the total combination.

HCF rule: Common primes, lowest exponents.

✍️ MCQ

Choose one

In the

HCF

rule, which exponents do we select for the common prime factors?

LCM rule: All primes, highest exponents.

✍️ T/F

True or False?

To find the

LCM

, we only consider the prime factors that are common to all numbers. True or false?

Difference 1: Which primes to include

HCF uses only COMMON primes.

Why? Because the HCF must DIVIDE both numbers.

If a prime (say $7$ ) is absent from one number (say $156$ ), then the HCF cannot contain $7$ — otherwise it would not divide $156$ .

✍️ Yes/No

Yes or No?

If we are finding the HCF of

12

and

15

, and we know

15

is not divisible by

2

, can the HCF contain the prime factor

2

?

LCM uses ALL primes.

Why? Because both numbers must DIVIDE the LCM.

If a prime (say $7$ ) appears in one number (say $126 = 2 \times 3^{2} \times 7$ ), then the LCM must contain $7$ — otherwise $126$ would not divide the LCM.

✍️ Yes/No

Yes or No?

To find the LCM of

10

(

2 \times 5

) and

14

(

2 \times 7

), must we include the prime factor

7

in our LCM?

Difference 2: Which exponent to choose.

HCF uses LOWEST exponents.

Why? The HCF must divide both numbers. If one number has $2^{1}$ and the other has $2^{2}$ , the HCF can have at most $2^{1}$ . Taking $2^{2}$ would mean the HCF does not divide the first number.

✍️ MCQ

Choose one

If we are finding the HCF of

2^3 \times 3^2

and

2^5 \times 3^1

, which power of

2

should we pick?

LCM uses HIGHEST exponents.

Why? Both numbers must divide the LCM. If one number has $2^{1}$ and the other has $2^{2}$ , the LCM must have at least $2^{2}$ . Taking only $2^{1}$ would mean the second number does not divide the LCM.

✍️ T/F

True or False?

To ensure that both original numbers can divide into the LCM, we must choose the highest power for every prime factor included.

Memory aid for HCF:

HCF must FIT INTO both numbers.
This means it must be small.
So, we use lowest exponents and common primes only.

Memory aid for LCM:

Both numbers must FIT INTO the LCM.
This means it must be big.
So, we use highest exponents and all primes involved.

2. Computing LCM of Two Numbers

Putting the LCM Rule into Action

Remember the key steps for finding the LCM:

List ALL primes from both factorisations (including those appearing in only one number).
Take the highest exponent for each prime.

To find the LCM, we need to include every prime factor at its maximum power.

📋 Given Info

Given Information:

$126 = 2 \times 3^{2} \times 7$
$156 = 2^{2} \times 3 \times 13$

✍️ Question

Find $L C M (126, 156)$

List all distinct primes, show the exponent comparison, and compute the final answer.

$126 = 2 \times 3^{2} \times 7$
$156 = 2^{2} \times 3 \times 13$

For LCM, you need ALL primes from both numbers — not just the common ones.

✍️ Yes/No

Yes or No?

If only one number has the prime factor

5

, do we include

5

in the LCM?

Let me show the systematic approach:

✍️ MCQ

Choose one

Which primes must be included in the LCM of

(2^2 \times 3^1)

and

(5^1)

?

Step 1: List all distinct primes from EITHER factorisation.

$126 = 2 \times 3^{2} \times 7$ → primes: {2, 3, 7}

$156 = 2^{2} \times 3 \times 13$ → primes: {2, 3, 13}

✍️ MCQ

Choose one

Which prime factor is present in

156

(

2^2 \times 3 \times 13

) but NOT in

126

(

2 \times 3^2 \times 7

)?

Combine: {2, 3, 7, 13} — four distinct primes total.

✍️ Yes/No

Yes or No?

Does the

LCM

require us to include every unique prime base found in either number?

Step 2: For each prime, find the HIGHEST exponent.

Now that we have our master list of primes (2, 3, 7, and 13), we need to decide how many of each to include. To ensure the LCM is large enough for both numbers to divide into it, we apply our second golden rule: Always pick the highest exponent for every prime.

✍️ MCQ

Choose one

For the prime factor

2

, we have

2^1

in

126

and

2^2

in

156

. Which exponent should we choose for the LCM?

Let's look at the full comparison table for our two numbers:

Prime	In 126	In 156	Max Exponent
2	1	2	2
3	2	1	2
7	1	0	1
13	0	1	1

Note: When a prime is absent from a number (like 7 is missing from 156), its exponent is effectively 0. Since the maximum of 1 and 0 is 1, we still include that prime in our LCM.

✍️ FIB

Fill in the blank

What is the maximum exponent for the prime factor

3

?

Step 3: Multiply.

$LCM = 2^{2} \times 3^{2} \times 7 \times 13$

✍️ MCQ

Choose one

What is the value of

2^2 \times 3^2

?

$= 4 \times 9 \times 7 \times 13$

$= 36 \times 7 \times 13$

$= 252 \times 13$

✍️ FIB

Fill in the blank

What is

252 \times 13

?

$= 3276$

Verification: Is our answer correct?

Does 126 divide 3276?

$3276 \div 126 = 26$

Result: Yes, it divides exactly!

✍️ Yes/No

Yes or No?

If

3276 \div 126

had a remainder, would 3276 still be a common multiple?

Now, let's check the second number:

Does 156 divide 3276?

$3276 \div 156 = 21$

Conclusion: Both exact. ✓ Correct.

A Common Mistake: Forgetting Non-Common Primes

If you had missed the non-common primes (7 and 13) and computed only $2^{2} \times 3^{2} = 36$ , check: does 126 divide 36?

✍️ Yes/No

Yes or No?

Can

126

divide into

36

?

No ( $126 > 36$ ). The LCM must be $\geq$ both numbers.

✍️ Yes/No

Yes or No?

If the LCM of

10

and

20

is

5

, is that correct?

3. Verifying with the Product Property

You've been working with HCF and LCM using prime factorisation. Now here's something really useful:

For any two numbers $a$ and $b$ , there's a beautiful relationship:

$HCF (a, b) \times LCM (a, b) = a \times b$

This gives you a way to check both answers at once. If the product doesn't match, at least one of your answers is wrong.

✍️ Question

Verification Challenge 🔍

For the numbers 612 and 1314, a student has computed:

HCF = $18$
LCM = $44676$

Verify these answers using the product property:

$HCF (a, b) \times LCM (a, b) = a \times b$

Show the computation on both sides.

The Product Property

The product property says: for any two positive integers $a$ and $b$ ,

$HCF (a, b) \times LCM (a, b) = a \times b$

✍️ Yes/No

Yes or No?

If

a = 6

and

b = 4

, does

\text{HCF}(6, 4) \times \text{LCM}(6, 4) = 24

?

This is a powerful verification tool. Let me show you how to use it.

✍️ MCQ

Choose one

Given

\text{HCF} = 5

,

\text{LCM} = 30

, and

a = 10

, find

b

.

Left side: HCF × LCM

$18 \times 44676$

To make this big multiplication easier, we can use the distributive property to break it down into parts.

✍️ MCQ

Choose one

Choose the best way to break down

44676

for easier multiplication:

Break it down: $18 \times 44676 = 18 \times 44000 + 18 \times 676$

$= 792000 + 12168$

✍️ FIB

Fill in the blank

792000 + 12168 = \text{?}

Final Result: $= 804168$

Right side: $a \times b$

Now, let's calculate the product of our original two numbers:

$612 \times 1314$

✍️ MCQ

Choose one

Which is a helpful way to split

1314

for multiplication?

Break it down:

$612 \times 1314 = 612 \times 1000 + 612 \times 314$ $= 612000 + 612 \times 300 + 612 \times 14$

✍️ FIB

Fill in the blank

Calculate

612 \times 300

.

$= 612000 + 183600 + 8568$ $= 804168$

Both sides = 804168. Match! ✓

This confirms that HCF = 18 and LCM = 44676 are both correct.

What if they did NOT match?

✍️ Yes/No

Yes or No?

If the two sides do not match, is the property itself wrong?

Then at least one of your answers is wrong — go back and check your factorisations and your min/max choices.

✍️ MCQ

Choose one

Where should you look first if your verification fails?

A Critical Limitation

Important caveat: This property works for TWO numbers only.

For three or more numbers, $HCF \times LCM$ does NOT equal the product of the numbers.

✍️ Yes/No

Yes or No?

Does

\text{HCF}(a, b, c) \times \text{LCM}(a, b, c) = a \times b \times c

?

How to Verify for Three Numbers

Since the product rule doesn't work, you verify three-number problems using the definitions themselves:

Check the HCF: Ensure the HCF divides all three numbers exactly.
Check the LCM: Ensure all three numbers divide the LCM exactly.

✍️ MCQ

Choose one

How do you verify the LCM of

4

,

5

, and

10

?

4. Complete Two-Number HCF and LCM from Scratch

Let's do a complete problem from scratch 📝

We're going to factorise two numbers, find their HCF, find their LCM, and verify our answers.

This is the full exam procedure — exactly what you'll need to do in your assessments.

✍️ Question

Practice Problem

Find $HCF (504, 540)$ and $LCM (504, 540)$

Show all your working:

Prime factorise both numbers
Find the HCF
Find the LCM
Verify using the product property:

$HCF (a, b) \times LCM (a, b) = a \times b$

Let's work through the HCF and LCM of 504 and 540 completely.

Factorise 504:

$504 \div 2 = 252$ $252 \div 2 = 126$

✍️ FIB

Fill in the blank

What is

126 \div 2

?

$126 \div 2 = 63$ $63 \div 3 = 21$

✍️ MCQ

Choose one

What is the prime factorisation of

21

?

$21 \div 3 = 7$ $7 \div 7 = 1$

The Result: $504 = 2^{3} \times 3^{2} \times 7$

Check: $8 \times 9 \times 7 = 72 \times 7 = 504$ ✓

Factorising 540

Now, let's apply the same logic to our second number, 540.

$540 \div 2 = 270$ $270 \div 2 = 135$

✍️ Yes/No

Yes or No?

Is

135

divisible by

2

?

Since 135 is odd, we move to the next prime: 3.

$135 \div 3 = 45$ $45 \div 3 = 15$ $15 \div 3 = 5$

✍️ FIB

Fill in the blank

What prime number divides

5

exactly?

$5 \div 5 = 1$

The Prime Factorisation of 540:

$540 = 2^{2} \times 3^{3} \times 5$

Check: $4 \times 27 \times 5 = 108 \times 5 = 540$ ✓

Comparing the Primes

Now that we have the prime factorisations for both numbers, let's put them side-by-side to find our HCF and LCM.

Comparison table:

Prime	504	540	Min (HCF)	Max (LCM)
2	3	2	2	3
3	2	3	2	3
5	0	1	—	1
7	1	0	—	1

✍️ MCQ

Choose one

Which primes are common to both

504

and

540

?

Finding the HCF

To find the HCF, we only look at the common primes and take their lowest (minimum) exponents.

HCF: Common primes are 2 and 3 (both present).

$HCF = 2^{2} \times 3^{2} = 4 \times 9 = 36$

✍️ MCQ

Choose one

Which exponent of

5

is used for the HCF if the exponents are

2

and

0

?

LCM: All primes: 2, 3, 5, 7.

$LCM = 2^{3} \times 3^{3} \times 5 \times 7 = 8 \times 27 \times 5 \times 7$ $= 216 \times 5 \times 7 = 1080 \times 7 = 7560$

✍️ MCQ

Choose one

In the LCM calculation, why did we use

2^3

instead of

2^2

?

Verification:

$HCF \times LCM = 36 \times 7560 = 272160$

✍️ FIB

Fill in the blank

If the property holds, what must be the product of

504 \times 540

?

$a \times b = 504 \times 540 = 272160$

Match! ✓ Both answers confirmed.

Key Observations from our Calculation

Notice the pattern in how we picked our primes:

For HCF: We used only 2 and 3 (the common primes), with their lowest exponents (2 and 2).
For LCM: We used all four primes (2, 3, 5, and 7), with their highest exponents (3, 3, 1, and 1).

✍️ Yes/No

Yes or No?

When calculating the LCM, do we only look at the prime factors that are common to both numbers?

Understanding the Magnitude

Now, look at the results we got:

HCF (36) is much smaller than both 504 and 540.
LCM (7560) is much larger than both numbers.

This makes sense because the HCF is a factor (it divides into them), while the LCM is a multiple (they divide into it).

✍️ Yes/No

Yes or No?

Can the HCF of two different positive integers ever be larger than their LCM?

The Fundamental Relationship

Finally, always remember this powerful rule for any two numbers $a$ and $b$ :

$HCF (a, b) \times LCM (a, b) = a \times b$

This property is your best friend for verifying your work or finding one value if you know the other three.

LCM of Two Numbers by Prime Factorisation

1. Contrasting HCF and LCM Rules

Difference 1: Which primes to include

Difference 2: Which exponent to choose.

2. Computing LCM of Two Numbers

Putting the LCM Rule into Action

Given Information:

Find LCM(126,156)LCM(126, 156)LCM(126,156)

Step 1: List all distinct primes from EITHER factorisation.

Step 2: For each prime, find the HIGHEST exponent.

Verification: Is our answer correct?

A Common Mistake: Forgetting Non-Common Primes

3. Verifying with the Product Property

Verification Challenge 🔍

The Product Property

Left side: HCF × LCM

Right side: a×ba \times ba×b

What if they did NOT match?

A Critical Limitation

How to Verify for Three Numbers

4. Complete Two-Number HCF and LCM from Scratch

Let's do a complete problem from scratch 📝

Practice Problem

Let's work through the HCF and LCM of 504 and 540 completely.

Factorising 540

Comparing the Primes

Finding the HCF

Key Observations from our Calculation

Understanding the Magnitude

The Fundamental Relationship

Find $L C M (126, 156)$

Right side: $a \times b$