You already know what HCF means:

The Highest Common Factor of two numbers is the largest number that divides both.

In CL-02, you found HCF using Euclid's Division Algorithm — repeated division without factorisation.

Now you will learn a second method that reads the HCF directly from the prime factorisations.

Method	Approach
Previous	Repeated Division
Current	Prime Factorisation

The rule is elegant:

Take the common primes with the lowest exponents.

In this section, you will:

Understand why this rule works
Practise it until it becomes automatic

1. Identifying Common Primes from Two Factorisations

Finding HCF by Prime Factorisation 🔢

The first step in finding HCF by prime factorisation is identifying which primes the two numbers share.

Common Mistake: Students often include primes that appear in only one number.

Let's make sure you can sort primes into two categories:

Common (Shared by both numbers)
Not Common (Appears in only one number)

📋 Given Info

Here are two numbers that have been factorised:

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

✍️ Question

Your turn 🤔

List all the prime factors of $126$ and all the prime factors of $156$ .

Which primes appear in BOTH factorisations (common primes)?
Which primes appear in only one?

Let me show you how to identify common primes.

✍️ Yes/No

Yes or No?

If a prime number is a factor of one number but not the other, is it a 'common prime'?

A 'common prime' is a prime that appears in BOTH factorisations — not just one.

✍️ MCQ

Choose one

Which primes appear in the factorisations of both

126

and

156

?

Let's check each prime systematically to see if they meet our definition of a 'common prime'.

Prime 2:

In $126 = 2 \times 3^{2} \times 7$ ? Yes (exponent 1).
In $156 = 2^{2} \times 3 \times 13$ ? Yes (exponent 2).
Verdict: Common (appears in both).

✍️ Yes/No

Yes or No?

Is the prime 3 a common prime for both

126

and

156

?

Prime 3:

In 126? Yes (exponent 2).
In 156? Yes (exponent 1).
Verdict: Common.

✍️ Yes/No

Yes or No?

Does the prime

7

appear in the prime factorisation of

156

?

Prime 7:

In 126? Yes (exponent 1).
In 156? No (7 does not appear at all).
Verdict: Not common (missing from 156).

Prime 13:

In 126? No.
In 156? Yes (exponent 1).
Verdict: Not common (missing from 126).

So the common primes are 2 and 3. The non-common primes are 7 (only in 126) and 13 (only in 156).

✍️ Yes/No

Yes or No?

If we want a number that divides both

126

and

156

, can we include

7

in our selection?

For HCF, we will only use the common primes. Non-common primes are excluded because the HCF must divide both numbers — and a prime absent from one number cannot divide that number.

✍️ MCQ

Choose one

Why do we ignore the prime

13

when calculating the HCF?

2. Applying the Lowest-Exponent Rule

You know which primes are common between two numbers. The next step is the exponent rule:

For each common prime, take the LOWEST of the two exponents.

Watch out! This is where students most often make errors—especially by accidentally using the highest exponent (which gives LCM, not HCF).

📋 Given Info

Here's what we have:

$126 = 2^{1} \times 3^{2} \times 7$
$156 = 2^{2} \times 3^{1} \times 13$

The common primes are 2 and 3.

Common Prime	Exponent in $126$	Exponent in $156$
2	$1$	$2$
3	$2$	$1$

We are tracking the exponents for each shared prime factor to prepare for the exponent rule.

✍️ Question

Using the common primes and the HCF rule (lowest exponents), compute:

HCF(126, 156)

Show which exponent you choose for each common prime.
Briefly explain why you chose those specific exponents.

The HCF Rule

The HCF rule says: for each common prime, take the LOWEST (smallest) exponent.

✍️ FIB

Fill in the blank

To find the HCF of two numbers, we must select the _________ exponent for each common prime factor.

Let's apply this to our common primes:

✍️ MCQ

Choose one

For the common prime

3

, should we pick exponent

1

or exponent

2

for the HCF?

Applying the Rule to Prime 2

Prime 2:

Exponent in $126$ : 1 (since $126 = 2^{1} \times 3^{2} \times 7$ )
Exponent in $156$ : 2 (since $156 = 2^{2} \times 3 \times 13$ )
Lowest of 1 and 2: 1
So we take: $2^{1}$

✍️ MCQ

Choose one

Which of these is the lower power of

2

?

Applying the Rule to Prime 3

Prime 3:

Exponent in $126$ : 2 (since $126 = 2 \times 3^{2} \times 7$ )
Exponent in $156$ : 1 (since $156 = 2^{2} \times 3^{1} \times 13$ )
Lowest of 2 and 1: 1
So we take: $3^{1}$

✍️ FIB

Fill in the blank

What is

2^1 \times 3^1

?

The Final HCF

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

So, the Highest Common Factor of $126$ and $156$ is $6$ .

Why lowest?

Because the HCF must divide BOTH numbers.

If we took $2^{2}$ instead of $2^{1}$ , the HCF would contain $4$ as a factor. But $126$ only has one factor of $2$ ( $126 = 2 \times 63$ ). So $4$ does not divide $126$ . The HCF would be wrong.

✍️ Yes/No

Yes or No?

If we used

2^2 = 4

in our HCF calculation, would

4

be a factor of

126

?

Similarly, if we took $3^{2}$ instead of $3^{1}$ , the HCF would contain $9$ as a factor.

But $156$ only has one factor of $3$ ( $156 = 2^{2} \times 3 \times 13$ ). So $9$ does not divide $156$ .

✍️ MCQ

Choose one

Why can't we use

3^2

in the HCF of

126

and

156

?

The lowest exponent is the safe choice — it guarantees the HCF divides both numbers.

Verification

Now that we've calculated the HCF as $2^{1} \times 3^{1} = 6$ , let's verify it actually works. Remember, an HCF must divide both numbers perfectly.

Quick check: Does 6 divide 126?

$126 \div 6 = 21$

Yes! ✓

✍️ MCQ

Choose one

What is

156 \div 6

?

Does 6 divide 156?

$156 \div 6 = 26$

Yes! ✓

Correct! Since 6 divides both numbers perfectly, we have successfully found the HCF.

3. Full HCF Computation from Scratch

Let's put it all together! 🎯

You've learned the pieces — now it's time to combine them:

Factorise two numbers into primes
Identify which primes are common to both
Apply the lowest-exponent rule
Compute the HCF

This is the complete skill as tested on the board exam.

✍️ Question

Find HCF(612, 1314) using the prime factorisation method.

Show your complete working:

(a) The prime factorisation of each number
(b) Which primes are common to both
(c) The HCF computation using lowest exponents

Let me walk through the complete procedure for HCF(612, 1314).

Step 1: Factorise 612.

612 is even: $612 \div 2 = 306$ . 306 is even: $306 \div 2 = 153$ .

✍️ FIB

Fill in the blank

What is the sum of the digits of

153

?

153 is odd. Digit sum: $1 + 5 + 3 = 9$ , divisible by 3. $153 \div 3 = 51$ . 51: digit sum $5 + 1 = 6$ , divisible by 3. $51 \div 3 = 17$ . 17 is prime. Done.

✍️ MCQ

Choose one

Is

17

a prime number or a composite number?

Combining everything together, we get:

$612 = 2 \times 2 \times 3 \times 3 \times 17 = 2^{2} \times 3^{2} \times 17$

Verify: $4 \times 9 \times 17 = 36 \times 17 = 612$ . ✓ Correct.

Step 2: Factorise 1314.

1314 is even: $1314 \div 2 = 657$

✍️ FIB

Fill in the blank

What is the sum of the digits of

657

?

657 is odd. Digit sum: $6 + 5 + 7 = 18$ , which is divisible by 3.

$657 \div 3 = 219$

✍️ Yes/No

Yes or No?

Is

219

divisible by

3

?

219: digit sum $2 + 1 + 9 = 12$ , divisible by 3. $219 \div 3 = 73$

73 is prime (not divisible by 2, 3, 5, 7; and $8^{2} = 64 < 73 < 81 = 9^{2}$ , so we only need to check up to 8).

So, the complete factorisation is:

$1314 = 2 \times 3 \times 3 \times 73 = 2 \times 3^{2} \times 73$

Verify: $2 \times 9 \times 73 = 18 \times 73 = 1314$ . ✓ Correct.

Step 3: Identify common primes.

612: prime factors are 2, 3, 17.

✍️ MCQ

Choose one

Which prime numbers are factors of

1314

?

1314: prime factors are 2, 3, 73.

Common Primes: 2 (in both) and 3 (in both).

✍️ Yes/No

Yes or No?

Is

73

a common prime factor for both

612

and

1314

?

Not common:

17 (only in 612)
73 (only in 1314)

Step 4: Apply lowest exponents.

For 2: exponents are 2 (in 612) and 1 (in 1314). Min = 1.

✍️ MCQ

Choose one

Between

2^2

and

2^1

, which one has the lowest exponent?

For 3: exponents are 2 (in 612) and 2 (in 1314). Min = 2.

Note: when both exponents are equal (like $3^{2}$ in both), the min is that same value.

✍️ FIB

Fill in the blank

If we have

3^2

in both numbers, what power of

3

do we include in the HCF?

$HCF = 2^{1} \times 3^{2} = 2 \times 9 = 18$

Verify: $612 \div 18 = 34$ (exact). $1314 \div 18 = 73$ (exact). Both divide evenly. ✓ Correct.

⚠️ Common pitfall: Don't stop too early!

When factorising large numbers, it is very easy to mistake certain numbers for prime numbers and stop your work there.

✍️ MCQ

Choose one

Is

51

a prime number or a composite number?

The Truth about 51 and 219:

$51 = 3 \times 17$ (NOT prime)
$219 = 3 \times 73$ (NOT prime)

Both of these can be further broken down into smaller prime factors.

✍️ MCQ

Choose one

If you stop your factorisation at

51

or

219

, what will be the result?

If you stopped your factorisation at $51$ or $219$ , the exponential form would be wrong and the HCF would be incorrect.

4. Explaining Why the Rule Works

Being able to execute the procedure is important, but understanding WHY it works prevents errors under pressure.

If you forget whether to use lowest or highest exponents, you can reason it out from the definition of HCF.

Let me show you a student's work and see if you can spot what went wrong.

📋 Given Info

A student computes HCF(36, 48) as follows:

$36 = 2^{2} \times 3^{2}$
$48 = 2^{4} \times 3$

The student writes: $HCF = 2^{4} \times 3^{2} = 144$

✍️ Question

The student calculated: $HCF (36, 48) = 144$

Analyze the error: Without computing the correct answer first, explain two ways you can immediately tell this answer is wrong.
Find the solution: After identifying the mistakes, determine the correct HCF.

Let me show you how to spot and fix this error.

Red flag #1: The 'HCF' is bigger than both numbers.

$144 > 48 > 36$ . But the HCF must DIVIDE both numbers. Can $144$ divide $36$ ? No — $144$ is bigger than $36$ .

✍️ Yes/No

Yes or No?

Can the HCF of two numbers ever be larger than the numbers themselves?

A number cannot be divided by something bigger than itself (with a whole-number result). So $144$ cannot possibly be a factor of $36$ , let alone a common factor of both.

Red flag #2: Direct check fails.

$36 \div 144 = 0.25$

Not a whole number. So $144$ is not a factor of $36$ at all.

✍️ FIB

Fill in the blank

If a number

x

is a factor of

36

, what should the remainder be when

36

is divided by

x

?

The error: The student used the HIGHEST exponents ( $2^{4}$ and $3^{2}$ ) instead of the LOWEST.

✍️ MCQ

Choose one

To find the HCF, do we use the highest or the lowest exponents?

Using the highest exponents gives a number that is TOO BIG to divide both original numbers.

The Correct Computation

Let's apply the rule properly for the numbers 36 and 48.

First, we look at their prime factorisations:

$36 = 2^{2} \times 3^{2}$ $48 = 2^{4} \times 3^{1}$

✍️ MCQ

Choose one

Which exponent of

2

should be used for the HCF?

For HCF, use LOWEST exponents of common primes:

Prime 2: exponents 2 and 4. Lowest = 2. Take $2^{2}$ .
Prime 3: exponents 2 and 1. Lowest = 1. Take $3^{1}$ .

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

✍️ FIB

Fill in the blank

What is

48 \div 12

?

Verify the Result

$36 \div 12 = 3$ ✓
$48 \div 12 = 4$ ✓

Both divisions are exact. Also, notice that $12 \leq 36$ and $12 \leq 48$ . This makes perfect sense!

Why we use Lowest Exponents for HCF

The conceptual key: The HCF must FIT INTO both numbers.

A big number cannot fit into a small number. So the HCF must be small — which is why we take the LOWEST exponents.

✍️ MCQ

Choose one

To find a number that fits INTO both

12

and

18

, would you look for a common factor or a common multiple?

Why we use Highest Exponents for LCM

The LCM, on the other hand, must be big enough for both numbers to fit into it — which is why LCM uses HIGHEST exponents.

✍️ MCQ

Choose one

Which exponents are used to calculate the LCM?

HCF of Two Numbers by Prime Factorisation

1. Identifying Common Primes from Two Factorisations

Finding HCF by Prime Factorisation 🔢

2. Applying the Lowest-Exponent Rule

The HCF Rule

Applying the Rule to Prime 2

Applying the Rule to Prime 3

The Final HCF

Why lowest?

Verification

3. Full HCF Computation from Scratch

Let's put it all together! 🎯

Step 2: Factorise 1314.

Step 3: Identify common primes.

Not common:

4. Explaining Why the Rule Works

The Correct Computation

Verify the Result

Why we use Lowest Exponents for HCF

Why we use Highest Exponents for LCM