Real Numbers Important Questions.

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1. Q1Easy

   Which of the following is an example of an irrational number?

   • A.√4
   • B.√9
   • C.√3
   • D.√16
   Answer: √3

   Explanation: √3 cannot be expressed as a ratio of two integers, making it an irrational number.

2. Q2Easy

   Which of the following is an example of a terminating decimal?

   • A.1/3
   • B.1/7
   • C.1/2
   • D.1/11
   Answer: 1/2

   Explanation: The fraction 1/2 has a terminating decimal expansion because its denominator is 2, which is a prime factor of 2.

3. Q3Easy

   Which of the following is an example of a non-terminating repeating decimal?

   • A.1/2
   • B.1/4
   • C.1/3
   • D.1/8
   Answer: 1/3

   Explanation: The fraction 1/3 has a non-terminating repeating decimal expansion, which is 0.333...

4. Q4Easy

   According to the Fundamental Theorem of Arithmetic, which of the following statements is true?

   • A.Every composite number can have multiple unique prime factorizations.
   • B.Every composite number has a unique prime factorization (ignoring order).
   • C.Prime numbers have multiple factorizations.
   • D.Composite numbers cannot be factorized into primes.
   Answer: Every composite number has a unique prime factorization (ignoring order).

   Explanation: The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers, disregarding the order of factors.

5. Q5Medium

   According to the Fundamental Theorem of Arithmetic, which of the following statements is true?

   • A.Every composite number has multiple prime factorizations.
   • B.Every integer greater than 1 can be represented as a sum of prime numbers.
   • C.Every composite number has a unique prime factorization (ignoring order).
   • D.Prime numbers are not factorizable into smaller integers.
   Answer: Every composite number has a unique prime factorization (ignoring order).

   Explanation: The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number itself or can be represented as the product of prime numbers, and this representation is unique up to ordering.

6. Q6Medium

   What does the uniqueness part of the Fundamental Theorem of Arithmetic imply about the prime factors of a number like 36?

   • A.The prime factors can be any combination of primes.
   • B.The exponents of the prime factors can vary arbitrarily.
   • C.The prime factors of 36 are only 2 and 3, and their exponents are unique.
   • D.36 can be expressed as a sum of primes.
   Answer: The prime factors of 36 are only 2 and 3, and their exponents are unique.

   Explanation: The uniqueness part of the theorem means that 36 can only be expressed as 2² × 3², no other combination of primes will yield 36.

7. Q7Medium

   Which theorem is used to prove that √2 is irrational?

   • A.Euclid’s Division Algorithm.
   • B.Fundamental Theorem of Arithmetic.
   • C.Pythagorean Theorem.
   • D.Fermat’s Last Theorem.
   Answer: Fundamental Theorem of Arithmetic.

   Explanation: The proof of the irrationality of √2 relies on the uniqueness of prime factorization to show that √2 cannot be expressed as a ratio of two integers.

8. Q8Medium

   If a prime number p divides a², then by the Fundamental Theorem of Arithmetic, what can be concluded about p?

   • A.p divides a³ only.
   • B.p does not divide a.
   • C.p divides a.
   • D.p divides a² but not a.
   Answer: p divides a.

   Explanation: By the divisibility rule derived from the Fundamental Theorem of Arithmetic, if a prime p divides a², then p must also divide the original number a.

9. Q9Hard

   Which of the following is a correct application of the Fundamental Theorem of Arithmetic in proving the irrationality of √2?

   • A.√2 can be expressed as a ratio of two integers.
   • B.The prime factorization of √2 is unique.
   • C.Assume √2 is rational and derive a contradiction using prime factorization.
   • D.√2 is a composite number.
   Answer: Assume √2 is rational and derive a contradiction using prime factorization.

   Explanation: The proof of the irrationality of √2 uses contradiction by assuming √2 is rational and showing that it leads to a contradiction based on prime factorization.

10. Q10Hard

    What is the key step in the proof that √2 is irrational?

    • A.Assuming √2 is irrational and deriving a contradiction.
    • B.Assuming √2 is rational and deriving that a prime number divides both the numerator and denominator.
    • C.Using the Pythagorean theorem to show √2 is irrational.
    • D.Using prime factorization to show √2 is rational.
    Answer: Assuming √2 is rational and deriving that a prime number divides both the numerator and denominator.

    Explanation: The key step involves assuming √2 is rational and showing that it leads to a contradiction, implying √2 cannot be rational.

11. Q11Hard

    If 6ⁿ ends with the digit 0, what must be true about n?

    • A.n must be even
    • B.n must be greater than or equal to 1
    • C.n must be prime
    • D.n must be odd
    Answer: n must be greater than or equal to 1

    Explanation: For a number to end with a 0, it must be divisible by 10, which requires factors of both 2 and 5. Since 6 includes both, any natural number n will satisfy this condition.

12. Q12Hard

    Which of the following is a key step in proving that √2 is irrational?

    • A.Show that √2 can be expressed as a fraction with an integer numerator and denominator.
    • B.Assume √2 is rational and express it as a fraction in lowest terms, then derive a contradiction.
    • C.Use the Fundamental Theorem of Arithmetic to factorize √2 into primes.
    • D.Prove that √2 can be written as a terminating decimal.
    Answer: Assume √2 is rational and express it as a fraction in lowest terms, then derive a contradiction.

    Explanation: The proof by contradiction involves assuming √2 is rational, expressing it as a fraction, and then showing that this leads to a contradiction with the Fundamental Theorem of Arithmetic.

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