Consider the following statements regarding the classification and properties of numbers: … — State PSC Exam

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

I. The number $0$ is the smallest positive even integer, and it can be produced by calculating the sum of two distinct irrational numbers.
II. The rational fraction $\frac{22}{7}$ represents the exact value of the mathematical constant $\pi$ on the real number line.
III. Every prime number greater than $2$ is an odd integer, and the square of any odd integer always leaves a remainder of $1$ when divided by $8$ .

Which of the statements given above is/are correct?

Only IIIAnswer
B
Only I and III
C
Only II and III
D
I, II, and III

Answer

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

\pi

present in the other statements.

Step-by-Step Solution

Evaluate Statement I regarding the properties of the number

0

and the sum of irrationals.

Statement I is false.

While the sum of two distinct irrational numbers (e.g.,

\sqrt{2}

and

-\sqrt{2}

) can indeed be

0

, the number

0

itself is neither positive nor negative. Describing it as the 'smallest positive even integer' is incorrect. The smallest positive even integer is

2

Evaluate Statement II regarding the relationship between

\pi

and

\frac{22}{7}

Statement II is false.

The constant

\pi

is an irrational number (a non-terminating, non-repeating decimal), while

\frac{22}{7}

is a rational number. They are not exactly equal;

\frac{22}{7}

is merely a convenient fractional approximation used in basic calculations.

Evaluate Statement III regarding primes and the square of odd integers.

Statement III is true.

The only even prime is

2

, so all primes greater than

2

are odd. For any odd integer

n = 2k + 1

, its square is

(2k+1)^2 = 4k^2 + 4k + 1 = 4k(k+1) + 1

. Since

k(k+1)

is the product of two consecutive integers, it must be even (let

k(k+1) = 2m

). Thus,

4(2m) + 1 = 8m + 1

, proving it always leaves a remainder of

1

when divided by

8

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.

Key Concept

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

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