(Smallest common multiple of 8,12 and 15) - (Highest common factor of 64 and 80) is equal to

Options

A. 120
B. 130
C. 125
D. 104

Answer

To solve the question, we need to find the smallest common multiple (LCM) of 8, 12, and 15, and then subtract the highest common factor (HCF) of 64 and 80 from it.

### Step 1: Find the LCM of 8, 12, and 15

1. **Prime Factorization**:
- 8 = 2 × 2 × 2 = \(2^3\)
- 12 = 2 × 2 × 3 = \(2^2 \times 3^1\)
- 15 = 3 × 5 = \(3^1 \times 5^1\)

2. **Identify the highest power of each prime factor**:
- For 2: The highest power is \(2^3\) (from 8).
- For 3: The highest power is \(3^1\) (from both 12 and 15).
- For 5: The highest power is \(5^1\) (from 15).

3. **Calculate the LCM**:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]
- First, calculate \(8 \times 3 = 24\).
- Then, calculate \(24 \times 5 = 120\).

So, the LCM of 8, 12, and 15 is **120**.

### Step 2: Find the HCF of 64 and 80

1. **Prime Factorization**:
- 64 = 2 × 2 × 2 × 2 × 2 × 2 = \(2^6\)
- 80 = 2 × 2 × 2 × 2 × 5 = \(2^4 \times 5^1\)

2. **Identify the lowest power of each common prime factor**:
- For 2: The lowest power is \(2^4\) (from 80).

3. **Calculate the HCF**:
\[ \text{HCF} = 2^4 = 16 \]

So, the HCF of 64 and 80 is **16**.

### Step 3: Subtract the HCF from the LCM

Now, we subtract the HCF from the LCM:
\[ \text{Result} = \text{LCM} - \text{HCF} = 120 - 16 = 104 \]

Thus, the final answer is **104**.

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