Find the HCF of he numbers in each of the following , using the prime factorization method :
504, 980

Answer

To find the HCF (Highest Common Factor) of the numbers 504 and 980 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of 504
1. Start with the number 504.
2. Divide by the smallest prime number (2):
- 504 ÷ 2 = 252
3. Divide 252 by 2:
- 252 ÷ 2 = 126
4. Divide 126 by 2:
- 126 ÷ 2 = 63
5. Now, 63 is not divisible by 2, so we try the next prime number (3):
- 63 ÷ 3 = 21
6. Divide 21 by 3:
- 21 ÷ 3 = 7
7. Finally, 7 is a prime number, so we stop here.

The prime factorization of 504 is:
\[ 504 = 2^3 \times 3^2 \times 7^1 \]

### Step 2: Prime Factorization of 980
1. Start with the number 980.
2. Divide by the smallest prime number (2):
- 980 ÷ 2 = 490
3. Divide 490 by 2:
- 490 ÷ 2 = 245
4. Now, 245 is not divisible by 2, so we try the next prime number (5):
- 245 ÷ 5 = 49
5. Now, 49 is not divisible by 5, so we try the next prime number (7):
- 49 ÷ 7 = 7
6. Finally, divide 7 by 7:
- 7 ÷ 7 = 1

The prime factorization of 980 is:
\[ 980 = 2^2 \times 5^1 \times 7^2 \]

### Step 3: Identify Common Factors
Now, we will identify the common prime factors from both factorizations:
- From 504: \( 2^3, 3^2, 7^1 \)
- From 980: \( 2^2, 5^1, 7^2 \)

The common prime factors are:
- For 2: The minimum power is \( 2^2 \)
- For 7: The minimum power is \( 7^1 \)

### Step 4: Calculate the HCF
Now, we multiply the common prime factors:
\[ \text{HCF} = 2^2 \times 7^1 = 4 \times 7 = 28 \]

Thus, the HCF of 504 and 980 is **28**.

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