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Difficulty: HardClassification of Numbers

Analyze the following three mathematical assertions concerning the classification and properties of numbers:

Assertion I: The expression 32−16÷4×2\sqrt{32} - 16 \div 4 \times \sqrt{2} evaluates to an irrational number.
Assertion II: For any prime number pp that is strictly greater than 22, the value of p2−1p^2 - 1 is always a multiple of 88.
Assertion III: The number 00 is classified as a rational number and an even integer, but it is neither a prime nor a composite number.

Which of the assertions provided above are logically correct?

  1. Assertions II and III are the only correct statements.Answer
  2. B
    Assertions I, II, and III are all correct statements.
  3. C
    Assertion III is the only correct statement.
  4. D
    Assertions I and II are the only correct statements.

Answer

Assertions II and III are the only correct statements.
Assertion II is mathematically accurate because the square of any odd prime minus one always factors into 4k(k+1)4k(k+1), which is a multiple of 88. Assertion III is also accurate as zero strictly fits the definitions of a rational number (0/10/1) and an even integer (0=2×00 = 2 \times 0), while being neither prime nor composite. Assertion I is false because correctly evaluating the expression from left-to-right yields 00, which is a rational number.

Step-by-Step Solution

1
Evaluate Assertion I using the correct order of operations (BODMAS/PEMDAS).
32−16÷4×2=42−4×2=42−42=0\sqrt{32} - 16 \div 4 \times \sqrt{2} = 4\sqrt{2} - 4 \times \sqrt{2} = 4\sqrt{2} - 4\sqrt{2} = 0.
Division and multiplication have equal precedence and are evaluated from left to right. The result, 00, is a rational number, making Assertion I false.
2
Analyze Assertion II for primes p>2p > 2.
All primes p>2p > 2 are odd, so p=2k+1p = 2k + 1 for some integer kk. Then p2−1=(2k+1)2−1=4k2+4k=4k(k+1)p^2 - 1 = (2k+1)^2 - 1 = 4k^2 + 4k = 4k(k+1).
Since kk and k+1k+1 are consecutive integers, one must be even. Thus, k(k+1)k(k+1) is an even integer, making 4k(k+1)4k(k+1) a multiple of 88. Assertion II is strictly true.
3
Verify the classifications of the number 00 in Assertion III.
00 can be written as 0/10/1 (rational), is divisible by 22 without remainder (even), and has no strictly positive divisors that satisfy the definition of primes or composites.
By standard mathematical definitions, 00 is rational, even, and neither prime nor composite. Assertion III is true.

Key Concept

Classification of numbers involving rational/irrational properties, algebraic properties of prime numbers, and the precise classification of zero.

Alternative Method

For Assertion II, students under time pressure can test the first few qualifying prime numbers (e.g., p=3⇒32−1=8p=3 \Rightarrow 3^2-1=8; p=5⇒52−1=24p=5 \Rightarrow 5^2-1=24) to quickly establish confidence in the truth of the statement without a formal algebraic proof.
Estimated Time:2m 0s
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Topics

Classification of Numbers

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