Find the HCF of 1048 and 1441.

Options

A. 11
B. 311
C. 131
D. 113

Answer

To find the HCF (Highest Common Factor) of 1048 and 1441, we can use the method of prime factorization. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 1048
We start by dividing 1048 by the smallest prime number, which is 2.

1. **1048 ÷ 2 = 524**
2. **524 ÷ 2 = 262**
3. **262 ÷ 2 = 131** (131 is a prime number)

So, the prime factorization of 1048 is:
\[ 1048 = 2^3 \times 131^1 \]

### Step 2: Prime Factorization of 1441
Next, we perform prime factorization on 1441. We check divisibility by prime numbers:

1. **1441 is not divisible by 2** (it’s odd).
2. **1441 ÷ 3 = 480.33** (not divisible).
3. **1441 ÷ 5 = 288.2** (not divisible).
4. **1441 ÷ 7 = 205.857** (not divisible).
5. **1441 ÷ 11 = 131** (11 is a prime factor).

So, the prime factorization of 1441 is:
\[ 1441 = 11^1 \times 131^1 \]

### Step 3: Identify Common Factors
Now we compare the prime factorizations of both numbers:

- **1048 = 2^3 \times 131^1**
- **1441 = 11^1 \times 131^1**

The only common prime factor is **131**.

### Step 4: Determine the HCF
The HCF is the product of the lowest powers of the common prime factors. Here, the only common factor is 131.

Thus, the HCF of 1048 and 1441 is:
\[ \text{HCF} = 131 \]

### Final Answer
The HCF of 1048 and 1441 is **131**.

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