How do we directly find the HCF without listing all the factors?

Question

Homi · Accepted Answer

We can find the HCF by using prime factors.

Step 1 — Find prime factors

First, let us find prime factors of 60. We divide 60 by the smallest prime number. We keep dividing until we get 1.

$60 \div 2 = 30$

$30 \div 2 = 15$

$15 \div 3 = 5$

$5 \div 5 = 1$

So, the prime factors of 60 are:

$\boxed{60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3^1 \times 5^1}$

Next, let us find prime factors of 84. We divide 84 by the smallest prime number. We keep dividing until we get 1.

$84 \div 2 = 42$

$42 \div 2 = 21$

$21 \div 3 = 7$

$7 \div 7 = 1$

So, the prime factors of 84 are:

$\boxed{84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3^1 \times 7^1}$

Diagram 1

Step 2 — Identify common prime factors

Now, we compare the prime factors of 60 and 84. We look for prime factors present in both numbers. Both numbers have prime factor 2. Both numbers have prime factor 3. Number 60 has prime factor 5. Number 84 does not. Number 84 has prime factor 7. Number 60 does not. So, the common prime factors are 2 and 3.

Let us find the lowest power for each common factor. For prime factor 2, 60 has $2^2$ . For prime factor 2, 84 has $2^2$ . The lowest power of 2 is $2^2$ .

For prime factor 3, 60 has $3^1$ . For prime factor 3, 84 has $3^1$ . The lowest power of 3 is $3^1$ .

Step 3 — Multiply common prime factors

We multiply these common prime factors. We use their lowest powers. This product will be the HCF.

$HCF = 2^2 \times 3^1$

$= (2 \times 2) \times 3$

$= 4 \times 3$

$\boxed{\text{HCF} = \mathbf{12}}$

Answer

(i) We use the prime factorization method to find HCF. (ii) We find common prime factors with their lowest powers. (iii) The HCF of 60 and 84 is 12.

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