### Instruction: Alice chose five positive integers and found that their product was even. What is the maximum number of odd integers she could have chosen?### Response: To find the maximum number of odd integers Alice could have chosen, I need to think about what makes a product even.
A product is even if and only if it has at least one even factor.
So, to minimize the number of even factors, I want to have exactly one even factor and the rest odd.
Therefore, the maximum number of odd integers Alice could have chosen is 4, since she has to choose 5 integers in total.
I can verify this by picking any four odd integers and one even integer, say 1, 3, 5, 7, and 2.
Their product is 1 * 3 * 5 * 7 * 2, which is clearly even.