To determine which statement must be true about the four numbers Jenna multiplied together, we need to analyze the situation in which the result of multiplying four numbers and then dividing by -2 is positive.
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Understanding the result: When Jenna multiplied four numbers together and then divided by -2, the expression can be represented mathematically as: \[ \frac{a \times b \times c \times d}{-2} \] where \(a\), \(b\), \(c\), and \(d\) are the four numbers she multiplied.
For this result to be a positive value: \[ \frac{a \times b \times c \times d}{-2} > 0 \] This can be rearranged to: \[ a \times b \times c \times d < 0 \] Thus, the product of the four numbers must be negative.
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Determining the conditions for a negative product: The product of numbers is negative when there is an odd number of negative factors. This is because:
- A negative times a negative yields a positive, and
- A positive multiplied by a positive yields a positive. Therefore:
- If you multiply an even number of negative numbers, you get a positive product.
- If you multiply an odd number of negative numbers, you get a negative product.
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Applying to the options:
- None of the factors were negative: This cannot be true because if none of the factors were negative, the product would be positive.
- All of the factors were negative: This cannot be true because if all four factors were negative, their product would be positive.
- An odd number of factors were negative: Since the product is negative, this must be true. The only way for four factors to yield a negative product is if there is an odd number of negative factors.
- An even number of factors were negative: This cannot be true because it would yield a positive product.
Thus, the only statement that MUST be true is: An odd number of factors were negative.
2 answers