Which of the following numbers is divisible by 3, 4, 5 and 6?

Options

A. 36
B. 60
C. 80
D. 90

Answer

To determine which of the following numbers is divisible by 3, 4, 5, and 6, we need to find the Least Common Multiple (LCM) of these numbers. Here’s a step-by-step solution:

### Step 1: Identify the numbers
We are given the numbers: 3, 4, 5, and 6.

### Step 2: Prime Factorization
We will find the prime factorization of each number:
- **3**: \(3^1\)
- **4**: \(2^2\)
- **5**: \(5^1\)
- **6**: \(2^1 \times 3^1\)

### Step 3: Determine the highest power of each prime factor
Next, we identify the highest power of each prime factor from the factorizations:
- The highest power of **2** is \(2^2\) (from 4).
- The highest power of **3** is \(3^1\) (from 3 and 6).
- The highest power of **5** is \(5^1\) (from 5).

### Step 4: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 \]
Calculating this step-by-step:
- \(2^2 = 4\)
- \(3^1 = 3\)
- \(5^1 = 5\)

Now, multiply these results:
\[ 4 \times 3 = 12 \]
\[ 12 \times 5 = 60 \]

Thus, the LCM of 3, 4, 5, and 6 is **60**.

### Step 5: Conclusion
Any number that is divisible by 3, 4, 5, and 6 must be a multiple of 60. Therefore, the answer to the question is that **60** is the number that is divisible by 3, 4, 5, and 6.

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