To support the conjectures inductively, let us analyze and justify each statement one by one.

1. The product of an odd number and its consecutive number is an even number.

Inductive Proof:

• Base Case: Let the odd number be $n = 1$. Its consecutive number is $n + 1 = 2$. The product is: $n \cdot (n + 1) = 1 \cdot 2 = 2$ $2$ is even, so the base case holds.

• Inductive Step: Assume for $n$, the product $n \cdot (n + 1)$ is even. For the next odd number $n + 2$, its consecutive number is $n + 3$. The product is: $(n + 2) \cdot (n + 3)$ Expanding: $(n + 2)(n + 3) = n^2 + 5n + 6$ Since $n^2 + 5n$ (odd number multiplied and summed) is odd, and adding 6 (even number) makes the entire product even. Thus, the inductive step is proven.

Conclusion: The conjecture is true for all odd numbers.

2. Positive numbers whose last digit is zero are divisible by 10.

Inductive Proof:

• Base Case: Consider $n = 10$. The last digit is zero, and $10 \div 10 = 1$, so it is divisible by 10.

• Inductive Step: Assume $n$ is a positive number whose last digit is zero and divisible by 10. Then $n = 10k$, where $k$ is an integer. For $n + 10$: $n + 10 = 10k + 10 = 10(k + 1)$ This is clearly divisible by 10 since $k + 1$ is an integer. Hence, the inductive step is verified.

Conclusion: All positive numbers ending in zero are divisible by 10.

3. The product of an odd integer and an even integer is an odd integer.

This statement is incorrect because the product of an odd integer and an even integer is always an even integer, not an odd one. Here's why:

• Let the odd number be $2k + 1$, and let the even number be $2m$. Their product is: $(2k + 1) \cdot 2m = 4km + 2m = 2(2km + m)$ This is divisible by 2, making the product even.

Correction: The product of an odd integer and an even integer is an even integer.

Let me know if you'd like additional clarifications or further detailed steps! Here are related questions:

1. Why does the inductive proof work for generalizations about even and odd numbers?
2. What happens if we analyze these conjectures using counterexamples instead of induction?
3. How would the proof for divisibility by 10 change if we use modulo arithmetic?
4. Can induction prove non-numerical conjectures as effectively as numerical ones?
5. Why does the product of an odd and even number always result in an even number?

Tip: Always start with simple base cases and ensure your inductive step logically extends the conjecture.