If the product of 6 integers is negative, what is the maximum possible number of these integers that can be negative?

Introduction:
This is a sign and parity reasoning question related to the product of integers. It tests your understanding of how a negative product arises from multiplying positive and negative numbers together. Specifically, for a product of 6 integers to be negative, the number of negative factors must be odd. The question asks for the maximum possible number of negative integers under this condition.

Given Data / Assumptions:

• There are exactly 6 integers in the product.
• The overall product is negative.
• We can freely choose how many are negative and how many are positive, but the sign of the total product must remain negative.
• Zero is not explicitly discussed; we assume the product is truly negative (not zero), so no factor is zero.

Concept / Approach:
When multiplying real numbers, the sign of the product depends on how many factors are negative. Each pair of negative factors contributes a positive effect because (-1) * (-1) = +1. Therefore, the overall sign is negative if and only if the number of negative factors is odd. Since there are 6 integers in total, we need to find the largest odd number less than or equal to 6.

Step-by-Step Solution:
Let k be the number of negative integers among the 6.The sign of the product is (-1)^k times the product of absolute values.For the total product to be negative, (-1)^k must be -1.This happens exactly when k is odd (1, 3, 5, etc.).We must also have 0 ≤ k ≤ 6 because there are 6 integers in total.Among the allowed odd values 1, 3, 5, the maximum is 5.So at most 5 of the integers can be negative while still keeping the product negative.

Verification / Alternative check:
Example with 5 negatives: (-1) * (-2) * (-3) * (-4) * (-5) * (6). There are 5 negatives and 1 positive. Pair four of the negatives to get positive contributions and one leftover negative, so the overall sign is negative. With 6 negatives, such as (-1) * (-2) * (-3) * (-4) * (-5) * (-6), the product is positive because the exponent of -1 is 6, an even number.

Why Other Options Are Wrong:
1, 3: These are odd and would give a negative product, but they are not the maximum possible count of negative integers.2 and 4: These are even, and a product with an even number of negative factors is positive, not negative.

Common Pitfalls:
Thinking that any number of negatives will work as long as there is at least one negative, which is not true.Forgetting that six negatives produce a positive product.Confusing maximum possible count with the minimum or just any valid count.

Final Answer:
5