https://file.aiquickdraw.com/mathbot/file/1727397869841mh5g1vu7.jpg

Let's analyze the statements from the image and determine whether they are true or false, and provide counterexamples where necessary.

1. **The product of any two prime numbers is always odd.**
   - **False**. A counterexample would be the prime numbers 2 and 3. \(2 \times 3 = 6\), which is even. Therefore, the product of two prime numbers is not always odd.

2. **If two angles are complementary, then both angles must be acute.**
   - **True**. Complementary angles add up to 90°. Since any angle greater than or equal to 90° would make the sum exceed 90°, both angles must be less than 90°, i.e., acute.

3. **The square of a number is always larger than the number.**
   - **False**. A counterexample is the number 1. \(1^2 = 1\), which is not larger than 1. Another counterexample is any number between 0 and 1. For instance, \( (0.5)^2 = 0.25 \), which is smaller than 0.5.

4. **Two lines always intersect at a point.**
   - **False**. Parallel lines do not intersect at any point. Therefore, the statement is false.

Let me know if you'd like more detailed explanations or have any questions!

Here are 5 related questions:
1. Can the product of any two prime numbers be odd under certain conditions?
2. What is the sum of two complementary angles if one of them is \(35^\circ\)?
3. Can the square of a negative number ever be smaller than the number itself?
4. What is the geometric condition for two lines to be parallel?
5. How can you prove that the sum of complementary angles must always be less than 90°?

**Tip:** A counterexample is a powerful tool in proving a statement false. It requires just one specific case to invalidate a conjecture!

Determine if the conjecture is true or false. If false, provide a counterexample.

Analyze and determine the truth of various mathematical conjectures, including whether the product of two prime numbers is always odd, if complementary angles must be acute, the properties of squared numbers, and if two lines always intersect. Suitable for Grades 6-8.

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