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.