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Zorluk: ZorClassification of Numbers

Consider the following statements regarding the classification and properties of numbers:

I. The number 00 is the smallest positive even integer, and it can be produced by calculating the sum of two distinct irrational numbers.
II. The rational fraction 227\frac{22}{7} represents the exact value of the mathematical constant π\pi on the real number line.
III. Every prime number greater than 22 is an odd integer, and the square of any odd integer always leaves a remainder of 11 when divided by 88.

Which of the statements given above is/are correct?

  1. Only IIICevap
  2. B
    Only I and III
  3. C
    Only II and III
  4. D
    I, II, and III

Cevap

Only Statement III is correct. Statement I fails because 0 is not positive, and Statement II fails because pi is irrational and not exactly equal to 22/7.
The correct answer accurately isolates Statement III as the only mathematically true proposition. It correctly identifies the parity of prime numbers and the modulo 8 behavior of odd squares, while dismissing the flawed definitions of zero and π\pi present in the other statements.

Adım Adım Çözüm

1
Evaluate Statement I regarding the properties of the number 00 and the sum of irrationals.
Statement I is false.
While the sum of two distinct irrational numbers (e.g., 2\sqrt{2} and −2-\sqrt{2}) can indeed be 00, the number 00 itself is neither positive nor negative. Describing it as the 'smallest positive even integer' is incorrect. The smallest positive even integer is 22.
2
Evaluate Statement II regarding the relationship between π\pi and 227\frac{22}{7}.
Statement II is false.
The constant π\pi is an irrational number (a non-terminating, non-repeating decimal), while 227\frac{22}{7} is a rational number. They are not exactly equal; 227\frac{22}{7} is merely a convenient fractional approximation used in basic calculations.
3
Evaluate Statement III regarding primes and the square of odd integers.
Statement III is true.
The only even prime is 22, so all primes greater than 22 are odd. For any odd integer n=2k+1n = 2k + 1, its square is (2k+1)2=4k2+4k+1=4k(k+1)+1(2k+1)^2 = 4k^2 + 4k + 1 = 4k(k+1) + 1. Since k(k+1)k(k+1) is the product of two consecutive integers, it must be even (let k(k+1)=2mk(k+1) = 2m). Thus, 4(2m)+1=8m+14(2m) + 1 = 8m + 1, proving it always leaves a remainder of 11 when divided by 88.
4
Determine the correct option based on the evaluations.
The option stating 'Only III' is the correct choice.
Since Statements I and II contain mathematical misconceptions, Statement III is the only accurate assertion.

Anahtar Kavram

Classification of numbers including rational versus irrational properties, integer parity, and fundamental prime characteristics.
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Konular

Classification of Numbers

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