Problem 8 · 2005 AMC 8
Easy
Number Theory
parity-rules
Suppose m and n are positive odd integers. Which of the following must also be an odd integer?
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Answer: E — 3mn.
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Hint 1 of 2
Don't plug in numbers and test — track only whether each piece is odd or even. The one rule that does all the work: odd + odd = even, and odd − odd = even.
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Hint 2 of 2
A product is odd only when every factor is odd; the moment a sum of two odds appears, it turns even. Scan for which choice avoids any 'odd + odd.'
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Approach: track parity, not values
- Replace each variable with its parity. Note 3 is odd, and odd×odd stays odd, so 3n, 3m, 3mn are all odd.
- (A) odd + odd = even. (B) odd − odd = even. (C) odd + odd = even. (D) inside is odd·odd + odd = odd + odd = even, and even² = even.
- (E) odd·odd·odd = odd — the only one with no addition of two odds to spoil it.
- The big idea: adding or subtracting two odds always makes an even; multiplying odds keeps them odd. Choices A–D each smuggle in an odd±odd, so only the pure product survives.
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