Problem 3 · 2023 Math Kangaroo
Medium
Number Theory
divisibility
The two integers m and n are positive and odd. Which of the following numbers is odd?
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Answer: A — \(m \cdot n + 2\)
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Hint 1 of 2
Odd times odd is odd; odd plus odd is even.
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Hint 2 of 2
Track the parity of each expression using m and n both odd.
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Approach: parity of products and sums
- With m, n odd: m·n is odd, and adding 2 keeps it odd, so (A) m·n+2 is odd.
- Check the rest: (B) (m+1)(n+1) is even·even = even; (C) m+n+2 is even; (D) m(n+1) is odd·even = even; (E) m+n is even.
- Only choice A is odd.
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