What is the least number that is divisible by all the numbers `1` to `10`

Options

A. `10`
B. `100`
C. `504`
D. `2520`

Explanation

To find the least number that is divisible by all the numbers from 1 to 10, we need to determine the Least Common Multiple (LCM) of these numbers. Here’s a step-by-step solution:

### Step 1: List the numbers from 1 to 10
The numbers we need to consider are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10

### Step 2: Prime Factorization of each number
We will find the prime factorization of each number:
- **1**: 1 (not a prime)
- **2**: 2
- **3**: 3
- **4**: 2 × 2 (or \(2^2\))
- **5**: 5
- **6**: 2 × 3
- **7**: 7
- **8**: 2 × 2 × 2 (or \(2^3\))
- **9**: 3 × 3 (or \(3^2\))
- **10**: 2 × 5

### Step 3: Identify the highest powers of each prime factor
Now, we need to take the highest power of each prime number that appears in the factorizations:
- **2**: Highest power is \(2^3\) (from 8)
- **3**: Highest power is \(3^2\) (from 9)
- **5**: Highest power is \(5^1\) (from 5 and 10)
- **7**: Highest power is \(7^1\) (from 7)

### Step 4: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 \times 7^1 \]

Calculating this step-by-step:
1. \(2^3 = 8\)
2. \(3^2 = 9\)
3. \(5^1 = 5\)
4. \(7^1 = 7\)

Now multiply these values:
- First, calculate \(8 \times 9 = 72\)
- Next, calculate \(72 \times 5 = 360\)
- Finally, calculate \(360 \times 7 = 2520\)

Thus, the least number that is divisible by all the numbers from 1 to 10 is **2520**.

### Final Answer:
The least number that is divisible by all the numbers from 1 to 10 is **2520**.

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