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Difficulty: MediumEven-Odd Properties and Sign Rules

Let mm and nn be negative integers such that m<nm < n and m−nm - n is an odd integer. Which of the following expressions must be a positive even integer?

  1. (m−n)2+1(m - n)^2 + 1Answer
  2. B
    m+n+1m + n + 1
  3. C
    (−1)n(m−n)(-1)^n (m - n)
  4. D
    m2+n2m^2 + n^2
  5. E
    (m−n)3(m - n)^3

Answer

(m−n)2+1(m - n)^2 + 1 must be a positive even integer.
Because m−nm - n is given as an odd integer and m<nm < n, m−nm - n is a negative odd integer. Squaring any negative odd integer gives a positive odd integer. Adding 11 to a positive odd integer yields a positive even integer, which guarantees the result is always positive and even.

Step-by-Step Solution

1
Analyze the parity and sign of m−nm - n
m−nm - n is a negative odd integer.
Since m<nm < n, m−n<0m - n < 0. The problem specifies that m−nm - n is odd.
2
Evaluate the expression (m−n)2(m - n)^2
(m−n)2(m - n)^2 is a positive odd integer.
Squaring any non-zero real number yields a positive result. Squaring an odd integer always yields an odd integer.
3
Add 1 to (m−n)2(m - n)^2
(m−n)2+1(m - n)^2 + 1 is an even integer greater than or equal to 2.
Adding 1 to a positive odd integer converts it into a positive even integer.

Key Concept

Parity rules under arithmetic operations and exponents with signed integers
Estimated Time:1m 15s
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Topics

Even-Odd Properties and Sign Rules

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