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Q: When is the product of a negative integer and a positive integer less than both of its factors?

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Yes. The product of a negative integer and a positive integer is a negative integer.

Always.

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No, if a negative integer is multiplied by a positive integer, the product is negative. However, if both of the integers are either positive or negative, the product is positive.

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That's always the case.

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A negative integer multiplied by a negative integer is always a positive integer product. -x * -y = xy

The smallest positive integer is 1. 1 is the multiplicative identity; ie anything times 1 is itself. The greatest negative integer is the most positive negative integer which is -1. Therefore the product of the greatest negative integer and the smallest positive integer is the greatest negative integer which is -1.

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