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Question

Difficulty: MediumProperties of Integers

Let xx be a negative even integer and yy be a negative odd integer. Which of the following expressions will always result in a positive odd integer?

  1. x×y+1x \times y + 1Answer
  2. B
    x+yx + y
  3. C
    x×yx \times y
  4. D
    ∣x∣+y|x| + y

Answer

x×y+1x \times y + 1
When multiplying two negative integers, the result is positive. When multiplying an even integer by an odd integer, the result is always even. Therefore, the product of xx and yy is a positive even integer. Adding 1 to any even integer results in an odd integer. Thus, the expression always yields a positive odd integer.

Step-by-Step Solution

1
Determine the sign and parity of the product x×yx \times y.
x×yx \times y is a positive even integer.
Multiplying two negative numbers yields a positive number. Multiplying an even number by any integer yields an even number.
2
Evaluate the result of adding 1 to the product.
x×y+1x \times y + 1 is a positive odd integer.
Adding 1 to any even integer changes its parity to odd. Adding 1 to a positive number keeps it positive.
3
Check the validity of the other provided expressions.
Only x×y+1x \times y + 1 guarantees a positive odd result.
Other expressions result in negative odd values, positive even values, or values that can fluctuate in sign based on specific magnitudes.

Key Concept

Parity and Sign Properties of Integers
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Topics

Properties of Integers

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