Find the greatest number which will divide 304 and 298 so as to leave a remainder 4 in each case.

To solve this problem, we need to find the greatest number \( x \) that divides both 304 and 298, leaving a remainder of 4 in each case.

### Step-by-step Solution:

1. **Adjust the numbers to account for the remainder:**

   If a number \( x \) divides 304 leaving a remainder of 4, then \( 304 - 4 = 300 \) must be divisible by \( x \).

   Similarly, if \( x \) divides 298 leaving a remainder of 4, then \( 298 - 4 = 294 \) must also be divisible by \( x \).

2. **Restate the problem:**

   We now need to find the greatest number \( x \) that divides both 300 and 294 exactly.

3. **Find the greatest common divisor (GCD):**

   The greatest number \( x \) that divides both 300 and 294 is their greatest common divisor (GCD). 

   Let's find the GCD of 300 and 294.

   - **Prime factorization:**
     - \( 300 = 2^2 \times 3 \times 5^2 \)
     - \( 294 = 2 \times 3^3 \times 7 \)

   - **Find common prime factors:**
     - Both numbers have the prime factors \( 2 \) and \( 3 \).
     - The lowest powers of the common factors are \( 2^1 \) and \( 3^1 \).

   - **Calculate the GCD:**
     \[
     \text{GCD}(300, 294) = 2^1 \times 3^1 = 2 \times 3 = 6
     \]

4. **Conclusion:**

   The greatest number \( x \) that divides both 304 and 298, leaving a remainder of 4 in each case, is **6**.

### Final Answer:
The greatest number which will divide 304 and 298 to leave a remainder of 4 in each case is **6**.

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Would you like more details, or do you have any other questions?

Here are 5 related questions to explore further:
1. How do you find the least common multiple (LCM) of two numbers?
2. What is the relationship between the GCD and LCM of two numbers?
3. How can you find the GCD of more than two numbers?
4. What are the properties of prime numbers and their importance in finding GCD?
5. How do we extend the concept of GCD to polynomials?

**Tip:** Remember that the GCD of two numbers is the largest number that divides both of them without leaving a remainder.

Discover how to find the greatest number which divides 304 and 298, leaving a remainder of 4 in each case. This problem involves number theory concepts such as greatest common divisor (GCD) and prime factorization. Suitable for Grades 7-9.

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