You have just mastered HCF for two numbers.

The three-number version is almost identical — same rule, same procedure.

The one critical difference:

Common now means common to ALL three numbers, not just two.

This sounds obvious, but it is where most errors happen.

A prime that appears in two of the three numbers but not the third must be excluded from the HCF.

In this section, you will practise the comparison table method:

Feature	Benefit
Visual Alignment	See all factors at once
Systematic	No factor left behind
Error-free	High accuracy

Let's get systematic.

1. Identifying Common Primes Across Three Numbers

You've just mastered finding the HCF of two numbers using prime factorisation. Now let's level up to three numbers!

The method is almost identical — same rule, same procedure.
There's one critical difference:

'Common' now means common to ALL THREE numbers, not just two.

The trickiest part of three-number HCF is correctly identifying common primes.

A prime present in two numbers but absent from the third is NOT common.

Let's check your understanding with a quick diagnostic.

📋 Given Info

Given Information

Three numbers have been factorised into their prime components:

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

Note the different prime bases ( $2, 3, 5,$ and $7$ ) used across these three values.

✍️ Question

Question 🤔

Which primes are common to ALL three numbers (appear in all three factorisations)?
Which primes appear in fewer than three?
Explain why prime 5 should NOT be included in the HCF.

Determining 'Common' Factors

For three numbers, 'common' means a prime must appear in ALL THREE factorisations — not just some.

✍️ Yes/No

Yes or No?

If the prime factor

3

is present in the factorisations of

108

and

120

, but NOT in

252

, is it a common factor for all three numbers?

Let's check each prime systematically.

Systematic Check: Prime 2

Prime 2:

In $108 = 2^{2} \times 3^{3}$ ? ✓ Yes (exponent $2$ )
In $120 = 2^{3} \times 3 \times 5$ ? ✓ Yes (exponent $3$ )
In $252 = 2^{2} \times 3^{2} \times 7$ ? ✓ Yes (exponent $2$ )

✍️ Yes/No

Yes or No?

Is the prime factor

2

common to all three numbers?

All three? Yes — common.

Systematic Check: Prime 3

Prime 3:

In $108$ ? ✓ Yes (exponent $3$ )
In $120$ ? ✓ Yes (exponent $1$ )
In $252$ ? ✓ Yes (exponent $2$ )

✍️ Yes/No

Yes or No?

Is the prime factor

3

common to all three numbers?

All three? Yes — common.

Prime 5:

In 108? ✗ No ( $108 = 2^{2} \times 3^{3}$ , no factor of 5)
In 120? ✓ Yes (exponent 1)
In 252? ✗ No ( $252 = 2^{2} \times 3^{2} \times 7$ , no factor of 5)
All three? No — NOT common (missing from 108 and 252).

✍️ MCQ

Choose one

Why is the prime factor

5

excluded from our common factors list?

Prime 7:

In 108? ✗ No
In 120? ✗ No
In 252? ✓ Yes (exponent 1)
All three? No — NOT common (missing from 108 and 120).

✍️ Yes/No

Yes or No?

Is the prime factor

7

common to all three numbers?

So only 2 and 3 are common to all three.

Why does this matter?

The HCF must divide ALL three numbers. If a number is going to be a common factor, it has to be a factor of every single number in our set.

✍️ Yes/No

Yes or No?

If a number

x

is the HCF of

108, 120,

and

252

, must

x

be a factor of

108

?

If the HCF contained $5$ , it would need to divide $108$ .

But $108 = 2^{2} \times 3^{3}$ has no factor of $5$ at all. So no number containing $5$ can divide $108$ . Including $5$ in the HCF would break the definition.

✍️ Yes/No

Yes or No?

If

108

does not have

5

as a prime factor, can any number that is a multiple of

5

divide

108

exactly?

2. Computing Three-Number HCF

Now let's compute the HCF using the comparison table.

You need to take the minimum exponent across all THREE numbers for each common prime.

📋 Given Info

Here are the prime factorisations we're working with:

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

Common primes: $2$ and $3$ (these appear in all three numbers).

✍️ Question

Compute $H C F (108, 120, 252)$

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

Step 1: Identify the common primes across all three factorisations.
Step 2: Compare the exponents for those common primes.
Step 3: Select the correct exponent to calculate the $H C F$ .

Finding the HCF of Three Numbers

Let me walk through the min-exponent computation for three numbers.

✍️ MCQ

Choose one

Which two prime factors are common to

108, 120,

and

252

?

We have two common primes: 2 and 3. For each, we need the MINIMUM exponent across ALL THREE numbers.

Number	Factorisation	Exponent of $2$	Exponent of $3$
$108$	$2^{2} \times 3^{3}$	$2$	$3$
$120$	$2^{3} \times 3^{1} \times 5^{1}$	$3$	$1$
$252$	$2^{2} \times 3^{2} \times 7^{1}$	$2$	$2$
HCF		min(2, 3, 2) = ?	min(3, 1, 2) = ?

✍️ FIB

Fill in the blank

What is the minimum exponent for the prime factor

3

?

Prime 2:

In 108: exponent = 2
In 120: exponent = 3
In 252: exponent = 2

Minimum of {2, 3, 2} = 2

So we take: $2^{2}$

✍️ FIB

Fill in the blank

What is the value of

2^2

?

Prime 3:

In 108: exponent = 3
In 120: exponent = 1
In 252: exponent = 2

Minimum of {3, 1, 2} = 1

So we take: $3^{1}$

✍️ Yes/No

Yes or No?

To find the HCF of three numbers, can we just pick the minimum exponent from any two of the numbers?

The key: you must compare ALL THREE exponents, not just two. The minimum of three numbers is the smallest of all three.

Calculating the Final HCF

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

✍️ MCQ

Choose one

If

12

is the HCF, what should happen when we divide

108, 120,

and

252

by

12

?

Verification:

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

✍️ Yes/No

Yes or No?

Could there be any number larger than

12

that divides

108, 120,

and

252

exactly?

The HCF divides all three numbers. And it is the HIGHEST such common factor — you cannot find a number larger than 12 that divides all three.

Why is $3^{1}$ the right choice for prime 3, even though 108 has $3^{3}$ and 252 has $3^{2}$ ?

Because 120 only has $3^{1}$ . The weakest link determines the limit.

✍️ MCQ

Choose one

In the context of HCF, what does the 'weakest link' refer to?

If we took $3^{2} = 9$ , then 9 would need to divide 120.

But $120 \div 9 = 13.33$ — not exact. So $3^{2}$ is too much for 120 to handle.

✍️ Yes/No

Yes or No?

If we used

3^3 = 27

as a factor for the HCF, would it divide

252

exactly?

3. Full Three-Number HCF from Scratch

Time to test the complete procedure! 🎯

You've mastered HCF for two numbers. Now let's see if you can apply the same method to three numbers.

The Process is Identical:

Factorise each number into primes.
Compare the prime factors across all numbers.
Extract common primes with the lowest exponents.

The Key Difference:
'Common' now means present in ALL THREE numbers, not just two of them.

✍️ Question

Find HCF(60, 84, 90)

Using the prime factorisation method:

Factorise all three numbers
Identify the primes that are common to ALL three
Compute the HCF

Show all your factorisations and working.

Let me work through HCF(60, 84, 90) step by step.

Factorise 60: $60 \div 2 = 30$ . $30 \div 2 = 15$ . $15 \div 3 = 5$ . $5 \div 5 = 1$ .

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

✍️ MCQ

Choose one

In the factorisation

60 = 2^2 \times 3 \times 5

, what is the exponent of the prime factor

2

?

Factorise 84: $84 \div 2 = 42$ . $42 \div 2 = 21$ . $21 \div 3 = 7$ . $7 \div 7 = 1$ .

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

✍️ Yes/No

Yes or No?

Is

5

a prime factor of

84

?

Factorise 90: $90 \div 2 = 45$ . $45 \div 3 = 15$ . $15 \div 3 = 5$ . $5 \div 5 = 1$ .

$90 = 2 \times 3^{2} \times 5$

✍️ MCQ

Choose one

Which prime factor of

90

is NOT a factor of

84

?

Building the Comparison Table

Now that we have the factorisations, let's organise them into a comparison table. This makes it much easier to spot which primes are common to all three numbers.

Prime	60	84	90	Common?
2	2	2	1	Yes

Look at the row for prime factor 2. In 60, the exponent is 2. In 84, it's also 2. But in 90, the exponent is 1.

✍️ FIB

Fill in the blank

In the factorisation of

90

, what is the exponent of the prime factor

2

?

Next, let's look at prime 3.

Prime	60	84	90	Common?
3	1	1	2	Yes

In 60, the exponent is 1. In 84, it's also 1. In 90, it's 2. Again, because it appears in all three factorisations, it is common.

Now, pay close attention to the primes that are NOT common. This is where the 'all or nothing' rule for three numbers applies.

Prime	60	84	90	Common?
5	1	0	1	No (0 in 84)
7	0	1	0	No (0 in 60, 90)

Prime 5 is in 60 and 90, but it's missing from 84. Prime 7 is only in 84. Neither of these can be part of our HCF.

✍️ MCQ

Choose one

Why is the prime factor

5

NOT considered common for

60, 84,

and

90

?

Critical point: 5 appears in 60 and 90, but NOT in 84. Since $84 = 2^{2} \times 3 \times 7$ , the number 84 has no factor of 5. So 5 is not common to all three, and it is excluded from the HCF.

✍️ Yes/No

Yes or No?

If the prime factor

5

is missing from even one of the three numbers, can it be used to calculate the HCF?

If you included 5, the 'HCF' would be $6 \times 5 = 30$ . Check: does 30 divide 84? $84 \div 30 = 2.8$

No! So 30 is not a common factor.

✍️ T/F

True or False?

A number can only be the HCF if it divides all the given numbers without a remainder.

Calculating the final HCF

HCF = $2^{\min (2, 2, 1)} \times 3^{\min (1, 1, 2)}$

✍️ MCQ

Choose one

Why is

1

the chosen exponent for prime factor

2

in the HCF?

$2^{1} \times 3^{1} = 2 \times 3 = 6$

Verification

$60 \div 6 = 10$ ✓
$84 \div 6 = 14$ ✓
$90 \div 6 = 15$ ✓

All divisions are exact. The HCF is correct!

✍️ Yes/No

Yes or No?

Can

12

be a common factor of

60, 84,

and

90

?

HCF of Three Numbers by Prime Factorisation

1. Identifying Common Primes Across Three Numbers

Given Information

Question 🤔

Determining 'Common' Factors

Systematic Check: Prime 2

Systematic Check: Prime 3

Why does this matter?

2. Computing Three-Number HCF

Compute HCF(108,120,252)HCF(108, 120, 252)HCF(108,120,252)

Finding the HCF of Three Numbers

Calculating the Final HCF

3. Full Three-Number HCF from Scratch

Time to test the complete procedure! 🎯

Find HCF(60, 84, 90)

Building the Comparison Table

Calculating the final HCF

Verification

Compute $H C F (108, 120, 252)$