Find HCF and LCM of 12, 18, 24 using prime factorization

Question

Find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of 12, 18, and 24 using the prime factorization method.

(NCERT Class 6)

Solution — Step by Step

Find the prime factorization of each number

$12 = 2 \times 2 \times 3 = 2^2 \times 3$

$18 = 2 \times 3 \times 3 = 2 \times 3^2$

$24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3$

Find the HCF

HCF = product of the smallest power of each common prime factor.

Common primes: 2 and 3.

Smallest power of 2: $2^1$ (from 18)
Smallest power of 3: $3^1$ (from 12 and 24)

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

Find the LCM

LCM = product of the highest power of each prime factor that appears in any number.

Highest power of 2: $2^3$ (from 24)
Highest power of 3: $3^2$ (from 18)

$\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = \mathbf{72}$

Why This Works

The HCF takes the minimum of each prime's exponent because the common factor must divide ALL numbers — so we can only use what is available in every number. The LCM takes the maximum because the common multiple must be divisible BY all numbers — so we need enough of each prime to cover the "greediest" number.

We can verify: 72 is divisible by 12 (72/12 = 6), by 18 (72/18 = 4), and by 24 (72/24 = 3). And 6 divides 12, 18, and 24 evenly.

Alternative Method — Division Method for HCF

Divide the larger number by the smaller. Then divide the divisor by the remainder. Repeat until remainder is 0. The last divisor is the HCF.

For 24 and 18: $24 = 18 \times 1 + 6$ , then $18 = 6 \times 3 + 0$ . HCF = 6. Now find HCF(6, 12) = 6.

💡 Expert Tip

Quick check: HCF $\times$ LCM = product of the two numbers — but this only works for two numbers, not three. For three numbers, use the prime factorization method. It is the most reliable approach for exams.

Common Mistake

⚠️ Common Mistake

Students swap the rules: they take the highest power for HCF and the lowest for LCM. This gives the wrong answer. Remember: HCF = take the Humble (smallest) powers. LCM = take the Largest powers. This mnemonic helps avoid the mix-up.

Question

Solution — Step by Step

Why This Works

Alternative Method — Division Method for HCF

Common Mistake

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