Quiz & Worksheet - Counterexamples in Math from Study.com.

Let’s solve each question one by one, with clear explanations.

---

Question 1: Provide a counterexample for the proposition: “All numbers are either prime or composite.”

✔ Correct Answer: 1

Explanation:
- A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.
- A composite number is a natural number greater than 1 that has more than two positive divisors.
- The number 1 is neither prime nor composite — it only has one positive divisor (itself), so it doesn’t meet the definition of either.
- Negative numbers like -2 or -1 are not considered in the standard definitions of prime/composite (which apply to positive integers ≥ 2).
- 4 is composite (divisible by 1, 2, 4), so it doesn’t disprove the statement.

Thus, 1 is the correct counterexample because it is a number that is neither prime nor composite.

---

Question 2: Which of the following is a counterexample to the statement: “All prime numbers are odd.”

✔ Correct Answer: 2

Explanation:
- The statement claims that every prime number is odd.
- But 2 is a prime number (only divisible by 1 and 2), and it is even.
- So, 2 directly contradicts the statement — making it a perfect counterexample.
- 1 is not prime.
- 3 is prime and odd — supports the statement.
- 4 is composite — irrelevant.

So, 2 is the correct answer.

---

Question 3: The number 22 is a counterexample for which of the following conditional statements?

✔ Correct Answer: “If a number is divisible by 2, then it is also divisible by 4.”

Explanation:
We need to find the statement that 22 breaks (i.e., makes false).

Let’s test each option:

1. “If a number is even, then it ends with 2,4,6,8 or 0.”
→ 22 is even and ends with 2 → ✔ True. Not a counterexample.

2. “If a number is divisible by 2, then it is even.”
→ This is logically true — being divisible by 2 *is* the definition of even. 22 is divisible by 2 and is even → ✔ True. Not a counterexample.

3. “If a number is divisible by 2, then it is also divisible by 4.”
→ 22 is divisible by 2 (22 ÷ 2 = 11), but not divisible by 4 (22 ÷ 4 = 5.5).
→ So this conditional statement is false for 22 → ✔ This is the counterexample!

4. “If a number is odd, then it is not divisible by 2.”
→ 22 is even, so this statement doesn’t even apply to 22. It’s vacuously true for 22. Not a counterexample.

Thus, only the third statement is falsified by 22, making it the correct answer.

---

✔ Final Answers:

1. 1
2. 2
3. If a number is divisible by 2, then it is also divisible by 4.

Let me know if you’d like further clarification!