Find the HCF of he numbers in each of the following , using the prime factorization method :
1197, 5320, 4389

Answer

To find the HCF (Highest Common Factor) of the numbers 1197, 5320, and 4389 using the prime factorization method, we can follow these steps:

### Step 1: Prime Factorization of 1197
1. Start with 1197. Check for divisibility by the smallest prime number, which is 3.
- 1197 ÷ 3 = 399
2. Now factor 399:
- 399 ÷ 3 = 133
3. Next, factor 133:
- 133 ÷ 7 = 19
4. Finally, factor 19:
- 19 ÷ 19 = 1

So, the prime factorization of 1197 is:
\[ 1197 = 3^2 \times 7^1 \times 19^1 \]

### Step 2: Prime Factorization of 5320
1. Start with 5320. Since the last digit is 0, it is divisible by 2.
- 5320 ÷ 2 = 266
2. Factor 266:
- 266 ÷ 2 = 133
3. Factor 133 (as done previously):
- 133 ÷ 7 = 19
4. Factor 19:
- 19 ÷ 19 = 1

So, the prime factorization of 5320 is:
\[ 5320 = 2^3 \times 5^1 \times 7^1 \times 19^1 \]

### Step 3: Prime Factorization of 4389
1. Start with 4389. Check for divisibility by 7:
- 4389 ÷ 7 = 627
2. Factor 627:
- 627 ÷ 3 = 209
3. Factor 209:
- 209 ÷ 19 = 11
4. Factor 11:
- 11 ÷ 11 = 1

So, the prime factorization of 4389 is:
\[ 4389 = 3^1 \times 7^1 \times 19^1 \times 11^1 \]

### Step 4: Identify Common Factors
Now, we will identify the common prime factors from the factorizations obtained:
- **Factors of 1197**: \( 3^2, 7^1, 19^1 \)
- **Factors of 5320**: \( 2^3, 5^1, 7^1, 19^1 \)
- **Factors of 4389**: \( 3^1, 7^1, 19^1, 11^1 \)

The common prime factors are:
- \( 7^1 \)
- \( 19^1 \)

### Step 5: Calculate the HCF
To find the HCF, we multiply the common factors:
\[ HCF = 7^1 \times 19^1 = 7 \times 19 = 133 \]

Thus, the HCF of the numbers 1197, 5320, and 4389 is **133**.

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