Problem 1 · 1993 AJHSME
Easy
Arithmetic & Operations
check-choices
Which pair of numbers does NOT have a product equal to 36?
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Answer: C — {1/2, −72}.
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Hint 1 of 2
You don't even have to multiply all of them — scan for anything different first. Four pairs are both-positive or both-negative; one pair has a positive and a negative.
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Hint 2 of 2
The sign rule decides it: a positive times a negative is always negative, so that pair can't make +36.
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Approach: spot the odd sign first, then confirm
- Before multiplying, look at signs. A product is positive only when the two numbers have the same sign (both + or both −). Four of the pairs match — but {1/2, −72} is one positive, one negative, so its product must be negative.
- Check it: (1/2)(−72) = −36, not 36. The odd pair is {1/2, −72}.
- Why this transfers: on a 'which one is different' problem, hunt for the lone exception by a quick property (here, sign) before grinding through every computation — the outlier usually announces itself.
Another way — multiply all five:
- (−4)(−9)=36, (−3)(−12)=36, (1)(36)=36, (3/2)(24)=36 — all equal 36, but (1/2)(−72)=−36. The pair that fails is {1/2, −72}.
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