What is the HCF of 132 and 176?

Options

A. 33
B. 66
C. 44
D. 22

Answer

To find the HCF (Highest Common Factor) of 132 and 176, we will follow these steps:

### Step 1: Prime Factorization of 132
- Start by dividing 132 by the smallest prime number, which is 2.
- 132 ÷ 2 = 66
- Now, divide 66 by 2 again.
- 66 ÷ 2 = 33
- Next, divide 33 by the next smallest prime number, which is 3.
- 33 ÷ 3 = 11
- Finally, 11 is a prime number.

So, the prime factorization of 132 is:
\[ 132 = 2^2 \times 3^1 \times 11^1 \]

### Step 2: Prime Factorization of 176
- Start by dividing 176 by 2.
- 176 ÷ 2 = 88
- Divide 88 by 2.
- 88 ÷ 2 = 44
- Divide 44 by 2.
- 44 ÷ 2 = 22
- Divide 22 by 2.
- 22 ÷ 2 = 11
- Finally, 11 is a prime number.

So, the prime factorization of 176 is:
\[ 176 = 2^4 \times 11^1 \]

### Step 3: Identify Common Factors
Now, we will identify the common prime factors from the factorizations:
- From 132: \( 2^2 \times 3^1 \times 11^1 \)
- From 176: \( 2^4 \times 11^1 \)

The common prime factors are:
- For 2: The minimum power is \( 2^2 \)
- For 11: The minimum power is \( 11^1 \)

### Step 4: Calculate the HCF
Now, we multiply the common prime factors:
\[ HCF = 2^2 \times 11^1 \]
Calculating this gives:
\[ HCF = 4 \times 11 = 44 \]

### Final Answer
Thus, the HCF of 132 and 176 is **44**.

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