Problem 7 · 2000 AMC 8
Medium
Arithmetic & Operations
number-systemscareful-counting
What is the minimum possible product of three different numbers of the set {−8, −6, −4, 0, 3, 5, 7}?
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Answer: B — −280.
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Hint 1 of 2
'Minimum' here means *most negative*, not smallest in size. First ask: which sign-combinations of three numbers even give a negative product?
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Hint 2 of 2
Negative product ⇒ an ODD number of negatives: either one negative or all three. To make it as far below zero as possible, you want the product's *size* as large as possible — so pair big numbers with big numbers.
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Approach: negative-and-large: count signs first, then maximize magnitude
- A product of three numbers is negative only with an odd count of negatives — so use exactly one negative, or all three.
- All three negatives: (−8)(−6)(−4) = −192. One negative, biggest magnitudes: take the most negative number and the two largest positives, (−8)(7)(5) = −280.
- −280 is further below zero, so the minimum is −280. (The 0 in the set would kill the product — never include it when you want a nonzero extreme.)
- The principle: for 'most negative product,' handle the SIGN and the SIZE separately — fix an odd number of negatives, then grab the largest-magnitude factors.
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