What is the highest common factor of 48, 120 and 240?

Options

A. 24
B. 12
C. 36
D. 40

Answer

To find the highest common factor (HCF) of the numbers 48, 120, and 240, we can follow these steps:

### Step 1: Prime Factorization of Each Number
- **For 48**:
- 48 can be divided by 2:
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 is a prime number.
- Therefore, the prime factorization of 48 is \( 2^4 \times 3^1 \).

- **For 120**:
- 120 can be divided by 2:
- 120 ÷ 2 = 60
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 can be divided by 3:
- 15 ÷ 3 = 5
- 5 is a prime number.
- Therefore, the prime factorization of 120 is \( 2^3 \times 3^1 \times 5^1 \).

- **For 240**:
- 240 can be divided by 2:
- 240 ÷ 2 = 120
- We already know the factorization of 120, which is \( 2^3 \times 3^1 \times 5^1 \).
- Therefore, 240 can be expressed as \( 2^4 \times 3^1 \times 5^1 \).

### Step 2: Identify Common Factors
Now we have:
- \( 48 = 2^4 \times 3^1 \)
- \( 120 = 2^3 \times 3^1 \times 5^1 \)
- \( 240 = 2^4 \times 3^1 \times 5^1 \)

### Step 3: Take the Lowest Power of Each Common Prime Factor
- For the prime factor **2**:
- The powers are \( 4, 3, 4 \). The lowest power is \( 2^3 \).

- For the prime factor **3**:
- The powers are \( 1, 1, 1 \). The lowest power is \( 3^1 \).

- For the prime factor **5**:
- The powers are \( 0, 1, 1 \) (since 48 has no factor of 5). The lowest power is \( 5^0 \) (which is 1 and does not contribute to the HCF).

### Step 4: Calculate the HCF
Now, we multiply the lowest powers of all common prime factors:
- HCF = \( 2^3 \times 3^1 \times 5^0 = 8 \times 3 \times 1 = 24 \).

### Conclusion
The highest common factor of 48, 120, and 240 is **24**.

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