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Question

Difficulty: MediumProperties of Integers

Which of the following statements about integers is always true?

  1. The difference between any integer and its absolute value is always an even integer.Answer
  2. B
    The product of any non-zero integer and its absolute value is always a positive integer.
  3. C
    Subtracting a negative integer from its absolute value always results in zero.
  4. D
    The sum of any negative integer and its absolute value is always a negative integer.

Answer

The statement claiming that the difference between any integer and its absolute value is always an even integer.
For any non-negative integer, the difference between the integer and its absolute value is 0, which is an even integer. For any negative integer, the difference evaluates to twice the integer (2x), which is a multiple of 2 and therefore also even. Thus, the statement holds true for all integers.

Step-by-Step Solution

1
Analyze the behavior of the expression (x - |x|) for non-negative integers.
If x is 0 or positive, |x| = x. Therefore, x - |x| = x - x = 0.
To test the property for the first distinct category of integers.
2
Analyze the behavior of the expression (x - |x|) for negative integers.
If x is negative, |x| = -x. Therefore, x - |x| = x - (-x) = x + x = 2x.
To test the property for the remaining category of integers.
3
Determine the parity (odd/even status) of the results from the previous steps.
The number 0 is an even integer. The expression 2x is inherently a multiple of 2, making it an even integer regardless of the value of x.
To conclude whether the difference is universally even.

Key Concept

Parity and Absolute Value Properties
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Topics

Properties of Integers

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