H.C.F. of 128, 288 and 160 is :

Options

A. 16
B. 24
C. 32
D. 48

Answer

To find the Highest Common Factor (H.C.F.) of the numbers 128, 288, and 160, we can follow these steps:

### Step 1: Find the Prime Factorization of Each Number

**For 128:**
- Divide by 2: 128 ÷ 2 = 64
- Divide by 2: 64 ÷ 2 = 32
- Divide by 2: 32 ÷ 2 = 16
- Divide by 2: 16 ÷ 2 = 8
- Divide by 2: 8 ÷ 2 = 4
- Divide by 2: 4 ÷ 2 = 2
- Divide by 2: 2 ÷ 2 = 1

So, the prime factorization of 128 is:
\[ 128 = 2^7 \]

**For 288:**
- Divide by 2: 288 ÷ 2 = 144
- Divide by 2: 144 ÷ 2 = 72
- Divide by 2: 72 ÷ 2 = 36
- Divide by 2: 36 ÷ 2 = 18
- Divide by 2: 18 ÷ 2 = 9
- Divide by 3: 9 ÷ 3 = 3
- Divide by 3: 3 ÷ 3 = 1

So, the prime factorization of 288 is:
\[ 288 = 2^5 \times 3^2 \]

**For 160:**
- Divide by 2: 160 ÷ 2 = 80
- Divide by 2: 80 ÷ 2 = 40
- Divide by 2: 40 ÷ 2 = 20
- Divide by 2: 20 ÷ 2 = 10
- Divide by 2: 10 ÷ 2 = 5
- Divide by 5: 5 ÷ 5 = 1

So, the prime factorization of 160 is:
\[ 160 = 2^5 \times 5^1 \]

### Step 2: Identify the Common Prime Factors

Now, we will identify the common prime factors from the factorizations:
- 128: \( 2^7 \)
- 288: \( 2^5 \times 3^2 \)
- 160: \( 2^5 \times 5^1 \)

The only common prime factor is \( 2 \).

### Step 3: Determine the Lowest Power of the Common Prime Factor

The lowest power of the common prime factor \( 2 \) is:
- From 128: \( 2^7 \)
- From 288: \( 2^5 \)
- From 160: \( 2^5 \)

The lowest power is \( 2^5 \).

### Step 4: Calculate the H.C.F.

Thus, the H.C.F. of 128, 288, and 160 is:
\[ H.C.F. = 2^5 = 32 \]

### Final Answer:
The H.C.F. of 128, 288, and 160 is **32**.

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