Problem 6 · 1987 AJHSME
Medium
Arithmetic & Operations
sign-rules
The smallest product one could obtain by multiplying two numbers in the set {−7, −5, −1, 1, 3} is
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Answer: B — −21.
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Hint 1 of 2
"Smallest" means farthest to the left on the number line — most negative, not closest to zero. What kind of factors give a negative product at all?
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Hint 2 of 2
A negative product needs exactly one negative factor. To push it as far below zero as possible, make both factors' sizes as large as you can.
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Approach: make a negative product with the biggest possible size
- Two negatives or two positives both give a positive product, so the smallest (most-negative) result must be a negative times a positive.
- To drive it as far below zero as possible, take the largest available sizes: the biggest positive is 3 and the most-negative is −7. 3 × (−7) = −21.
- Watch the trap: −7 × −5 = +35 is the largest, not the smallest — it's positive. "Smallest" rewards the deepest negative, which is −21.
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