HCF = product of common factors with lowest power. LCM = product of all with highest power.

Find LCM and HCF of 12, 15, 21 Using Prime Factorisation

Question

Find the HCF and LCM of 12, 15, and 21 using the prime factorisation method.

Solution — Step by Step

We break each number into its prime factors:

12 = 2^2 \times 3

15 = 3 \times 5

21 = 3 \times 7

HCF takes only the factors that appear in all three numbers, and uses the lowest power.

The only prime common to 12, 15, and 21 is 3 (with power 1 in all three).

\text{HCF}(12, 15, 21) = 3

LCM takes every prime that appears in any of the numbers, using the highest power.

Prime	Highest Power
2	$2^2$ (from 12)
3	$3^1$ (from any)
5	$5^1$ (from 15)
7	$7^1$ (from 21)

\text{LCM}(12, 15, 21) = 2^2 \times 3 \times 5 \times 7 = 4 \times 3 \times 5 \times 7 = 420

For two numbers, we have HCF × LCM = Product of the numbers. But for three numbers, this doesn’t hold directly — so we verify by checking divisibility instead.

Does 420 divide evenly by 12, 15, and 21?

$420 \div 12 = 35$ ✓
$420 \div 15 = 28$ ✓
$420 \div 21 = 20$ ✓

And 3 divides all three numbers evenly. We’re good.

HCF = 3, LCM = 420

Why This Works

Prime factorisation works because every integer has a unique prime factorisation — this is the Fundamental Theorem of Arithmetic. Since the factorisation is unique, we can precisely identify what’s “common” across numbers (HCF) and what covers all of them (LCM).

Think of HCF as the intersection of prime factors — only what’s shared by all. LCM is the union — you need at least this much to be divisible by each number. The “lowest power for HCF, highest power for LCM” rule follows directly from this logic.

This method scales cleanly to three or more numbers, which is why it’s the standard technique for board exams.

Alternative Method

Using pairwise HCF to find HCF of three numbers:

\text{HCF}(12, 15) = 3

\text{HCF}(3, 21) = 3

So $\text{HCF}(12, 15, 21) = 3$ . Same answer, slightly more steps. The prime factorisation method is faster when you’re already doing the factorisation for LCM anyway.

In CBSE boards, the examiner wants to see the prime factorisation written out explicitly — even if you can spot the answer mentally. Write each step for full marks.

Common Mistake

Taking the highest power for HCF and lowest for LCM — completely backwards.

Students sometimes mix up the rules: HCF uses the lowest common power, LCM uses the highest. A quick way to remember: HCF is the highest common factor, so it’s the biggest thing that divides into all numbers — that means you can only use what every number has (lowest shared power). LCM needs to be divisible by all numbers — so you need the full highest power of each prime.

In this problem: if you mistakenly took $2^2 \times 3 \times 5 \times 7 = 420$ as HCF, you should notice that 420 doesn’t divide into 12 — that’s your check that something went wrong.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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