What is the HCF of 64, 120 and 144?

Options

A. 4
B. 8
C. 12
D. 8640

Answer

To find the Highest Common Factor (HCF) of the numbers 64, 120, and 144, we will use the method of prime factorization. Here’s a step-by-step solution:

### Step 1: Prime Factorization of Each Number

**Prime Factorization of 64:**
- 64 is even, so we divide by 2:
- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1

So, the prime factorization of 64 is:
\[ 64 = 2^6 \]

**Prime Factorization of 120:**
- 120 is even, so we divide by 2:
- 120 ÷ 2 = 60
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15 (15 is odd, so we switch to the next prime number)
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1

So, the prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

**Prime Factorization of 144:**
- 144 is even, so we divide by 2:
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9 (9 is odd, so we switch to the next prime number)
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 144 is:
\[ 144 = 2^4 \times 3^2 \]

### Step 2: Identify Common Factors

Now, we will identify the common prime factors from the factorizations:
- **64:** \( 2^6 \)
- **120:** \( 2^3 \times 3^1 \times 5^1 \)
- **144:** \( 2^4 \times 3^2 \)

The common prime factor is \( 2 \).

### Step 3: Determine the Lowest Power of Common Factors

For the common factor \( 2 \):
- The lowest power of \( 2 \) among the three numbers is \( 2^3 \) (from 120).

### Step 4: Calculate the HCF

Thus, the HCF of 64, 120, and 144 is:
\[ HCF = 2^3 = 8 \]

### Final Answer

The HCF of 64, 120, and 144 is **8**.

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