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Difficulty: MediumClassification of Numbers

A mathematician is formally classifying a set of numbers and analyzing their properties. Evaluate the mathematical validity of the following three propositions:

I. The number 00 is an even integer, but it is considered neither positive nor negative.
II. The fraction 227\frac{22}{7} is an irrational number because it is commonly used to represent the constant π\pi.
III. The absolute difference between any two distinct odd prime numbers is always an even composite number.

Which of the propositions given above is/are mathematically correct?

  1. Only Proposition IAnswer
  2. B
    Propositions I and II
  3. C
    Propositions I and III
  4. D
    Propositions II and III

Answer

Only Proposition I is mathematically correct.
Proposition I is correct because 0 is an integer that is divisible by 2 (making it even) and is strictly the neutral boundary between positive and negative values. Proposition II is incorrect because 22/7 is expressed as a ratio of two integers, making it a rational number, despite being used as an approximation for pi. Proposition III is incorrect because the difference between the odd prime numbers 5 and 3 is exactly 2, which is an even prime number rather than a composite number.

Step-by-Step Solution

1
Evaluate Proposition I regarding the properties of the number 0.
The number 0 is an integer, is divisible by 2 without a remainder (making it even), and serves as the boundary between positive and negative numbers. Therefore, Proposition I is true.
To verify the fundamental classification and parity of zero.
2
Evaluate Proposition II regarding the classification of the fraction 22/7.
The value 22/7 is a ratio of two integers (22 and 7). Any number that can be expressed as a fraction of integers with a non-zero denominator is a rational number. Therefore, Proposition II is false.
To distinguish between a rational fraction and the actual irrational constant (pi) it often approximates.
3
Evaluate Proposition III regarding the absolute difference between distinct odd prime numbers.
Consider the distinct odd prime numbers 5 and 3. Their absolute difference is |5 - 3| = 2. Since 2 is a prime number, the difference is not always an even composite number. Therefore, Proposition III is false.
To test the universal claim with a basic mathematical counterexample.

Key Concept

Classification of Rational vs. Irrational Numbers, Prime Numbers, and Properties of Zero
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Topics

Classification of Numbers

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