Which of the following is divisible by 8,15 and 6 ?

Options

A. 60
B. 96
C. 180
D. 240

Explanation

To determine which of the given numbers is divisible by 8, 15, and 6, we will follow these steps:

### Step 1: Find the LCM of 8, 15, and 6
To find the least common multiple (LCM), we can use the prime factorization method.

- **Prime factorization of 8**:
- 8 = 2 × 2 × 2 = \(2^3\)

- **Prime factorization of 15**:
- 15 = 3 × 5 = \(3^1 × 5^1\)

- **Prime factorization of 6**:
- 6 = 2 × 3 = \(2^1 × 3^1\)

Now, we take the highest power of each prime number that appears in the factorizations:

- For 2: The highest power is \(2^3\) (from 8).
- For 3: The highest power is \(3^1\) (from both 15 and 6).
- For 5: The highest power is \(5^1\) (from 15).

Now, we can calculate the LCM:
\[ \text{LCM} = 2^3 × 3^1 × 5^1 = 8 × 3 × 5 \]
Calculating this step-by-step:
- \(8 × 3 = 24\)
- \(24 × 5 = 120\)

Thus, the LCM of 8, 15, and 6 is **120**.

### Step 2: Check which numbers are divisible by 120
Now we need to check which of the given numbers is divisible by 120. The numbers provided are 240, 16, 96, and 180.

- **240**:
- \(240 ÷ 120 = 2\) (Divisible)

- **16**:
- \(16 ÷ 120 = 0.1333\) (Not divisible)

- **96**:
- \(96 ÷ 120 = 0.8\) (Not divisible)

- **180**:
- \(180 ÷ 120 = 1.5\) (Not divisible)

### Conclusion
The only number that is divisible by 8, 15, and 6 is **240**.

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