The three-number LCM extension follows the same pattern as the three-number HCF you already learned:

Same comparison table
Same prime factorisation approach

But here is the twist for LCM:

Take the max exponent instead of the min
Include ALL primes instead of only common ones

Warning: The product property $H C F \times L C M = a \times b$ does NOT extend to three numbers.

Since that shortcut is off the table, you will need a different verification strategy to check your work in this section.

1. Three-Number LCM Computation

You have computed HCF(108, 120, 252) = 12 in the previous section.

Now let's compute the LCM using the same factorisations.

Remember: For LCM, include ALL primes and take the HIGHEST exponent.

📋 Given Info

Here are the prime factorisations:

$108 = 2^{2} \times 3^{3}$
$120 = 2^{3} \times 3 \times 5$
$252 = 2^{2} \times 3^{2} \times 7$

All distinct primes appearing: 2, 3, 5, 7

✍️ Question

Find LCM(108, 120, 252)

For each prime, show the exponents across all three numbers and which one you choose.

Numbers to consider: $108, 120, 252$
Prime factors identified: $2, 3, 5, 7$

Task: Identify the correct exponent for each prime to calculate the final Least Common Multiple.

For LCM of three numbers, you need ALL primes from ALL three factorisations, each with the HIGHEST exponent.

✍️ Yes/No

Yes or No?

When finding the LCM, do we only include primes that appear in all three numbers?

Let me build the full comparison table:

Prime	108	120	252	Max
2	2	3	2	3
3	3	1	2	3
5	0	1	0	1
7	0	0	1	1

✍️ FIB

Fill in the blank

Looking at the table, what is the highest exponent (Max) for the prime

3

?

Notice:

5 has exponent 0 in 108 and 252, but 1 in 120. Max = 1. So 5 IS included in LCM.
7 has exponent 0 in 108 and 120, but 1 in 252. Max = 1. So 7 IS included in LCM.

✍️ FIB

Fill in the blank

If a prime number has an exponent of

0

in two numbers but an exponent of

2

in the third number, what exponent do we take for the LCM?

This is the key difference from HCF:

For HCF: A prime with ANY zero is excluded (it must be common, or in ALL).
For LCM: A prime with ANY non-zero is included (it only needs to be in at least one).

✍️ T/F

True or False?

To be included in the HCF, a prime must appear in the factorisation of all three numbers.

Assembling the LCM

Based on the highest exponents we identified for each prime factor, our formula is:

LCM = $2^{3} \times 3^{3} \times 5 \times 7$

✍️ MCQ

Choose one

What is the value of

3^3

?

Compute Step by Step

Let's calculate the values of the powers first:

$2^{3} = 8$
$3^{3} = 27$

Now, let's start the multiplication:

$8 \times 27 = 216$

✍️ FIB

Fill in the blank

Calculate

216 \times 5

:

Final Calculation

Continuing the steps:

$216 \times 5 = 1080$
$1080 \times 7 =$ 7560

The LCM of 108, 120, and 252 is 7560.

Verification: Testing our Result

Now that we've calculated the LCM as 7560, it's a good habit to check our work.

Remember, the Least Common Multiple must be divisible by every number in the set. If our answer is correct, we should be able to divide 7560 by 108, 120, and 252 without any remainders.

✍️ Yes/No

Yes or No?

If we divide our LCM by

108

and get a remainder of

5

, would

7560

still be a correct common multiple?

The Math Check:

$7560 \div 108 = 70$
$7560 \div 120 = 63$
$7560 \div 252 = 30$

All results are whole numbers! This means the division is exact in every case.

✍️ T/F

True or False?

The LCM is divisible by all three numbers.

2. Why the Product Property Fails for Three Numbers

🔍 Quick Check Time!

You've already learned that for two numbers, there's a neat relationship:

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

Many students assume this extends to three numbers:

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

Let's test whether this assumption is actually correct!

📋 Given Info

Here's the situation:

For the numbers 108, 120, and 252, we have:

HCF = $12$
LCM = $7560$

A student wants to verify these answers by checking whether: $HCF \times LCM = 108 \times 120 \times 252$

✍️ Question

Your task:

Compute both sides:

Left side: $HCF \times LCM = 12 \times 7560 = ?$
Right side: $108 \times 120 \times 252 = ?$

Does this verification work? Why or why not?

This is a common trap. Let me clear it up.

The product property: $HCF (a, b) \times LCM (a, b) = a \times b$

✍️ Yes/No

Yes or No?

Was the product of HCF and LCM equal to the product of the three numbers in our example?

This is TRUE for exactly two numbers.

The textbook states this explicitly on p.11 with a caution:

'The above result is true for two numbers only.'

✍️ T/F

True or False?

The property

\text{HCF}(p, q, r) \times \text{LCM}(p, q, r) = p \times q \times r

is true.

Verifying the Product Property

Let's check for three numbers:

Left side: $HCF \times LCM = 12 \times 7560 = 90, 720$

✍️ FIB

Fill in the blank

The property

\text{HCF} \times \text{LCM} = \text{Product}

works for exactly ___ numbers.

Right side: $108 \times 120 \times 252$

$= 108 \times 120 = 12, 960$
$= 12, 960 \times 252 = 3, 265, 920$

✍️ Yes/No

Yes or No?

Is

90,720

equal to

3,265,920

?

$90, 720 \neq 3, 265, 920$ . Not even close.

So the product property fails for three numbers. This is not a calculation error — it is a genuine mathematical fact. The formula $HCF \times LCM = product$ works ONLY for two numbers.

How to verify three-number answers instead:

1. Check HCF: Does 12 divide all three?

$108 \div 12 = 9$ ✓
$120 \div 12 = 10$ ✓
$252 \div 12 = 21$ ✓

✍️ Yes/No

Yes or No?

If

24

also divides all three numbers, could

12

be the HCF?

2. Check LCM: Do all three divide 7560?

$7560 \div 108 = 70$ ✓
$7560 \div 120 = 63$ ✓
$7560 \div 252 = 30$ ✓

✍️ MCQ

Choose one

Which calculation verifies the LCM of

a, b,

and

c

?

3. Check 'highest' and 'lowest': Is there a common factor larger than 12? Is there a common multiple smaller than 7560? If you cannot find one after trying a few candidates, your answers are likely correct.

For the board exam, showing the divisibility checks is sufficient verification for three-number problems.

3. Complete Three-Number HCF and LCM from Scratch

Final Gate Challenge 🎯

This is the complete test of everything you've learned about HCF and LCM with three numbers.

To succeed, you will need to:

Factorise all three numbers
Build one comparison table
Read off both HCF and LCM from that table
Verify your answers

This is the exam-ready version of the full skill!

✍️ Question

Final Gate Challenge

Find both HCF and LCM of 36, 60, and 84 using a comparison table.

Show all prime factorisations.
Construct the comparison table.
State the HCF and LCM.
Verify both answers using the divisibility rules we discussed.

Remember: The product property $H C F \times L C M = a \times b \times c$ does not work for three numbers!

Let me do the complete procedure for 36, 60, and 84.

Factorise 36:

36 ÷ 2 = 18
18 ÷ 2 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1

So, $36 = 2^{2} \times 3^{2}$

✍️ MCQ

Choose one

In the prime factorisation of

36

, what are the exponents of the primes

2

and

3

?

Factorise 60:

60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1

So, $60 = 2^{2} \times 3 \times 5$

✍️ Yes/No

Yes or No?

Does the prime number

5

appear in the factorisation of

36

?

Factorise 84:

84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1

So, $84 = 2^{2} \times 3 \times 7$

✍️ MCQ

Choose one

Which prime factor is unique to the factorisation of

84

among

36, 60,

and

84

?

Building the Comparison Table

Now that we've found the prime factorisations, let's organise everything into a comparison table. This makes it much easier to pick the right exponents for our HCF and LCM.

Prime	36	60	84	Common?	Min	Max
2	2	2	2	Yes	2	2
3	2	1	1	Yes	1	2

✍️ MCQ

Choose one

Looking at the row for the prime

3

, what is the Minimum exponent?

Now, let's look at the primes that are not shared by everyone.

Prime	36	60	84	Common?	Min	Max
5	0	1	0	No	—	1
7	0	0	1	No	—	1

Note: For LCM, we don't care if a prime is common. We just need the highest exponent it has anywhere in the list!

✍️ Yes/No

Yes or No?

Is the prime number

7

common to all three numbers?

✍️ FIB

Fill in the blank

What is the Maximum exponent for the prime

7

?

Finding the HCF

To find the HCF (Highest Common Factor) of our three numbers, we only look at:

Common primes (those that appear in all three factorisations)
The minimum exponents for those common primes.

✍️ MCQ

Choose one

Which primes are common to

36, 60,

and

84

?

Since only 2 and 3 are common, we take their lowest powers from the table:

$HCF = 2^{2} \times 3^{1} = 4 \times 3 = 12$

✍️ Yes/No

Yes or No?

If we used the maximum exponent

3^2

, would the result still be a common factor?

Verification

Let's verify that 12 is indeed a factor of all three numbers:

$36 \div 12 = 3$ ✓
$60 \div 12 = 5$ ✓
$84 \div 12 = 7$ ✓

Since 12 divides all three numbers exactly, our HCF is correct!

LCM: All primes ( $2, 3, 5, 7$ ), max exponents.

Unlike the HCF we just found, for the LCM, we don't just look for what's common. We must include every single prime factor that appears in any of the numbers, and we always pick the maximum exponent for each.

✍️ Yes/No

Yes or No?

Do we include primes that are NOT common to all numbers when calculating the LCM?

LCM $= 2^{2} \times 3^{2} \times 5 \times 7$ $= 4 \times 9 \times 5 \times 7$ $= 36 \times 5 \times 7$ $= 180 \times 7$ $=$ 1260

✍️ FIB

Fill in the blank

What is

180 \times 7

?

Verify: $1260 \div 36 = 35, 1260 \div 60 = 21, 1260 \div 84 = 15$ . All exact. ✓

Just like we verified the HCF, we must check if our LCM is divisible by all our original numbers: $36, 60,$ and $84$ .

Key observations from this problem:

All three numbers have $2^{2}$ , so the min AND max for prime 2 are both 2.

✍️ MCQ

Choose one

If the minimum and maximum exponents are the same, what does it mean?

5 is in 60 only (excluded from HCF, included in LCM).
7 is in 84 only (excluded from HCF, included in LCM).

✍️ MCQ

Choose one

Why is 5 excluded from the HCF?

The product property does NOT apply:
- $12 \times 1260 = 15120$
- $36 \times 60 \times 84 = 181440$
- Not equal.

LCM of Three Numbers by Prime Factorisation

1. Three-Number LCM Computation

Find LCM(108, 120, 252)

Assembling the LCM

Compute Step by Step

Final Calculation

Verification: Testing our Result

2. Why the Product Property Fails for Three Numbers

Here's the situation:

Verifying the Product Property

How to verify three-number answers instead:

3. Complete Three-Number HCF and LCM from Scratch

Final Gate Challenge 🎯

Final Gate Challenge

Building the Comparison Table

Finding the HCF

Verification

Key observations from this problem: