Make a conjecture for each scenario. Show your work. 3. the square of an even number 4. the product of two odd numbers and a multiple of 2 Find one counterexample to show that each conjecture is false. 5. For two real numbers a and b, a is either equal to b or greater than b. 6. All quadrilaterals are parallelograms.

Conjecture: The square of an even number is always even. Work: Let's take an even number, say 2. The square of 2 is 4, which is even. Let's take another even number, say 4. The square of 4 is 16, which is also even. Therefore, we can conjecture that the square of an even number is always even.

Conjecture: The product of two odd numbers and a multiple of 2 is always even. Work: Let's take two odd numbers, say 3 and 5, and a multiple of 2, say 4. The product of 3, 5, and 4 is 60, which is even. Let's take another set of two odd numbers, say 7 and 9, and a multiple of 2, say 6. The product of 7, 9, and 6 is 378, which is also even. Therefore, we can conjecture that the product of two odd numbers and a multiple of 2 is always even.

Counterexample: Let's take two real numbers, say a = 2 and b = 3. Here, a is not equal to b and a is not greater than b. Therefore, the statement "For two real numbers a and b, a is either equal to b or greater than b" is false.

Counterexample: A rectangle is a quadrilateral but not a parallelogram. In a rectangle, opposite sides are parallel but adjacent sides are not. Therefore, the statement "All quadrilaterals are parallelograms" is false.

Conjecture: The square of an even number is always even. Work: Let's take an even number, say 2. The square of 2 is 4, which is even. Let's take another even number, say 4. The square of 4 is 16, which is also even. Therefore, we can conjecture that the square of an even number is always even.

Conjecture: The product of two odd numbers and a multiple of 2 is always even. Work: Let's take two odd numbers, say 3 and 5, and a multiple of 2, say 4. The product of 3, 5, and 4 is 60, which is even. Let's take another set of two odd numbers, say 7 and 9, and a multiple of 2, say 6. The product of 7, 9, and 6 is 378, which is also even. Therefore, we can conjecture that the product of two odd numbers and a multiple of 2 is always even.

Counterexample: Let's take two real numbers, say a = 2 and b = 3. Here, a is not equal to b and a is not greater than b. Therefore, the statement "For two real numbers a and b, a is either equal to b or greater than b" is false.

Counterexample: A rectangle is a quadrilateral but not a parallelogram. In a rectangle, the opposite sides are parallel but the adjacent sides are not. Therefore, the statement "All quadrilaterals are parallelograms" is false.