Find the HCF of 420 and 1782.

Answer

To find the HCF (Highest Common Factor) of 420 and 1782, we can follow these steps:

### Step 1: Prime Factorization of 420
1. Start by dividing 420 by the smallest prime number, which is 2.
- \( 420 \div 2 = 210 \)
2. Divide 210 by 2 again.
- \( 210 \div 2 = 105 \)
3. Now, divide 105 by the next smallest prime number, which is 3.
- \( 105 \div 3 = 35 \)
4. Next, divide 35 by the next prime number, which is 5.
- \( 35 \div 5 = 7 \)
5. Finally, 7 is a prime number, so we stop here.

The prime factorization of 420 is:
\[ 420 = 2^2 \times 3^1 \times 5^1 \times 7^1 \]

### Step 2: Prime Factorization of 1782
1. Start by dividing 1782 by 2.
- \( 1782 \div 2 = 891 \)
2. Now, divide 891 by the next smallest prime number, which is 3.
- \( 891 \div 3 = 297 \)
3. Divide 297 by 3 again.
- \( 297 \div 3 = 99 \)
4. Divide 99 by 3 once more.
- \( 99 \div 3 = 33 \)
5. Finally, divide 33 by 3.
- \( 33 \div 3 = 11 \)
6. 11 is a prime number, so we stop here.

The prime factorization of 1782 is:
\[ 1782 = 2^1 \times 3^4 \times 11^1 \]

### Step 3: Identify Common Factors
Now, we will identify the common prime factors from both factorizations:
- For \( 2 \): The minimum power is \( 2^1 \)
- For \( 3 \): The minimum power is \( 3^1 \)

### Step 4: Calculate HCF
To find the HCF, we multiply the common prime factors:
\[ \text{HCF} = 2^1 \times 3^1 = 2 \times 3 = 6 \]

Thus, the HCF of 420 and 1782 is **6**.

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