Divisibility Rules Sheet 4 - Practice identifying numbers divisible by 2 through 10.

Problem Analysis:

The task is to find a number between 300 and 400 that is divisible by all the numbers listed in the "Divisibility Rules Sheet." The numbers are:

- 2, 3, 4, 5, 6, 8, 9, 10

To solve this, we need to determine the least common multiple (LCM) of these numbers. A number divisible by all these numbers must be a multiple of their LCM.

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Step-by-Step Solution:

#### Step 1: Prime Factorization of Each Number
We start by finding the prime factorization of each number in the list:

1. 2: \( 2 \)
2. 3: \( 3 \)
3. 4: \( 2^2 \)
4. 5: \( 5 \)
5. 6: \( 2 \times 3 \)
6. 8: \( 2^3 \)
7. 9: \( 3^2 \)
8. 10: \( 2 \times 5 \)

#### Step 2: Determine the LCM
The LCM is found by taking 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 \( 3^2 \) (from 9).
- For 5: The highest power is \( 5 \) (from 5 or 10).

Thus, the LCM is:
\[
\text{LCM} = 2^3 \times 3^2 \times 5
\]

#### Step 3: Calculate the LCM
Now, compute the value of the LCM:
\[
2^3 = 8, \quad 3^2 = 9, \quad 5 = 5
\]
\[
\text{LCM} = 8 \times 9 \times 5
\]

First, calculate \( 8 \times 9 \):
\[
8 \times 9 = 72
\]

Next, multiply by 5:
\[
72 \times 5 = 360
\]

So, the LCM is:
\[
\text{LCM} = 360
\]

#### Step 4: Verify the Range
The problem asks for a number between 300 and 400 that is divisible by all the given numbers. Since the LCM is 360, and it lies within the range 300 to 400, it satisfies the condition.

#### Step 5: Confirm Divisibility
To ensure 360 is divisible by all the numbers (2, 3, 4, 5, 6, 8, 9, 10), we can quickly check:
- \( 360 \div 2 = 180 \) (integer)
- \( 360 \div 3 = 120 \) (integer)
- \( 360 \div 4 = 90 \) (integer)
- \( 360 \div 5 = 72 \) (integer)
- \( 360 \div 6 = 60 \) (integer)
- \( 360 \div 8 = 45 \) (integer)
- \( 360 \div 9 = 40 \) (integer)
- \( 360 \div 10 = 36 \) (integer)

All divisions result in integers, confirming that 360 is divisible by all the numbers.

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Final Answer:

The number between 300 and 400 that is divisible by all the numbers listed is:
\[
\boxed{360}
\]