Problem 7 · 1990 AJHSME
Medium
Arithmetic & Operations
sign-productmaximize
When three different numbers from the set {−3, −2, −1, 4, 5} are multiplied, the largest possible product is
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Answer: C — 30.
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Hint 1 of 2
For the product to be as big as possible it must first be *positive*. With negatives around, what makes a product positive — how many minus signs do you need?
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Hint 2 of 2
An even number of negatives gives a positive product. So the play is: grab TWO negatives (to cancel the signs) and make them the biggest negatives you have, then pair with the biggest positive.
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Approach: even count of negatives for a positive, then maximize size
- First make sure you can even be positive. One negative would leave the product negative (a loser); using *two* negatives flips the signs to positive. So pick exactly two of the three negative numbers.
- To make the positive product as large as possible, use the two with the biggest size: (−3)(−2) = 6. Then multiply by the largest positive, 5.
- 6 × 5 = 30. (Check the rivals: 4×5 with one negative is negative; (−3)(−2)(4)=24 < 30.)
- *Why this transfers:* in any 'largest product' with negatives, the count of minus signs decides the sign first — settle the sign, *then* chase the biggest absolute values.
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