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The question asks for a counterexample to the statement: 

"If \( x \) is an integer that is greater than 20 and composite, then \( x \) is evenly divisible by 3."

We need to find an integer \( x \) from the given options that:
1. Is greater than 20,
2. Is composite (not prime), and
3. Is **not divisible by 3**.

### Let's evaluate each option:

- **Option A: \( x = 23 \)**
  - 23 is greater than 20, but it is a prime number, not composite. So, it doesn't satisfy the composite condition.
  
- **Option B: \( x = 27 \)**
  - 27 is greater than 20, and it is composite (\( 27 = 3^3 \)).
  - It is divisible by 3, so it satisfies the statement. Not a counterexample.
  
- **Option C: \( x = 10 \)**
  - 10 is composite (\( 10 = 2 \times 5 \)) but not greater than 20. So, this does not fit the conditions.

- **Option D: \( x = 25 \)**
  - 25 is greater than 20, and it is composite (\( 25 = 5^2 \)).
  - 25 is **not divisible by 3**, so this is a counterexample.

### Conclusion:
The correct answer is **D: \( x = 25 \)**, as it satisfies the conditions for being a counterexample.

Would you like more details on the reasoning? Here are some related questions:

1. What does it mean for a number to be composite?
2. Why are composite numbers important in this context?
3. Can prime numbers be counterexamples in such statements?
4. How do you test if a number is divisible by 3?
5. Can you give other examples of composite numbers greater than 20 that are not divisible by 3?

**Tip**: A composite number has more than two distinct factors, unlike a prime number which has only 1 and itself as divisors.

Select the value for x that is a counterexample to the following statement: If x is an integer that is greater than 20 and composite, then x is evenly divisible by 3.

Identify a counterexample to the statement: 'If x is an integer greater than 20 and composite, then x is divisible by 3.' Explore key concepts like composite numbers, divisibility rules, and counterexamples. Ideal for students in Grades 7-9.

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