Find the HCF of the numbers in each of the following , using the prime factorization method :
144, 252, 630

Answer

To find the HCF (Highest Common Factor) of the numbers 144, 252, and 630 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of Each Number
1. **Prime Factorization of 144**:
- Divide by 2: \( 144 \div 2 = 72 \)
- Divide by 2: \( 72 \div 2 = 36 \)
- Divide by 2: \( 36 \div 2 = 18 \)
- Divide by 2: \( 18 \div 2 = 9 \)
- Divide by 3: \( 9 \div 3 = 3 \)
- Divide by 3: \( 3 \div 3 = 1 \)

So, the prime factorization of 144 is:
\[ 144 = 2^4 \times 3^2 \]

2. **Prime Factorization of 252**:
- Divide by 2: \( 252 \div 2 = 126 \)
- Divide by 2: \( 126 \div 2 = 63 \)
- Divide by 3: \( 63 \div 3 = 21 \)
- Divide by 3: \( 21 \div 3 = 7 \)
- Divide by 7: \( 7 \div 7 = 1 \)

So, the prime factorization of 252 is:
\[ 252 = 2^2 \times 3^2 \times 7^1 \]

3. **Prime Factorization of 630**:
- Divide by 2: \( 630 \div 2 = 315 \)
- Divide by 3: \( 315 \div 3 = 105 \)
- Divide by 3: \( 105 \div 3 = 35 \)
- Divide by 5: \( 35 \div 5 = 7 \)
- Divide by 7: \( 7 \div 7 = 1 \)

So, the prime factorization of 630 is:
\[ 630 = 2^1 \times 3^2 \times 5^1 \times 7^1 \]

### Step 2: Identify Common Factors
Now, we will identify the common prime factors from the factorizations:
- For 2: The minimum power is \( 2^1 \) (from 630).
- For 3: The minimum power is \( 3^2 \) (common in all three).
- For 5: Not common in all three.
- For 7: Not common in all three.

### Step 3: Calculate the HCF
Now we multiply the common prime factors with their minimum powers:
\[ \text{HCF} = 2^1 \times 3^2 = 2 \times 9 = 18 \]

Thus, the HCF of 144, 252, and 630 is **18**.

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