CUET Preparation Today
Consider the following statements (A) To obtain prime numbers less than 121, we have to reject all the multiples of 2, 3, 5 and 7. Which of the statement(s) given above is/are correct? Choose the correct answer from the options given below: |
(A), (B) and (D) only (B), (C) and (D) only (C), (D), and (A) only (A), (B) and (C) only |
(A), (B) and (D) only |
The correct answer is Option (1) → (A), (B) and (D) only Evaluation of Statements (A) To obtain prime numbers less than 121, we have to reject all the multiples of 2, 3, 5 and 7. Correct. According to the Sieve of Eratosthenes, to find all primes up to a number $n$, you only need to check multiples of primes up to $\sqrt{n}$. Since $\sqrt{121} = 11$, we must check multiples of primes less than 11, which are 2, 3, 5, and 7. (Note: 1 is neither prime nor composite and is also rejected). (B) Every composite number less than 121 is divisible by a prime number less than 11. Correct. This is a corollary to the rule above. If a number $n < 121$ is composite, it must have at least one prime factor $p \leq \sqrt{n}$. Since the maximum value of $n$ is 120, and $\sqrt{120} \approx 10.95$, every such composite number must be divisible by a prime less than 11 (i.e., 2, 3, 5, or 7). (C) 173 is not a prime number. Incorrect. To check if 173 is prime, we check divisibility by primes up to $\sqrt{173} \approx 13.15$ (primes: 2, 3, 5, 7, 11, 13). Not divisible by 2 (odd), 3 (sum of digits is 11), or 5 (ends in 3). $173 \div 7 = 24.7$ $173 \div 11 = 15.7$ $173 \div 13 = 13.3$ Since it is not divisible by any of these, 173 is a prime number. (D) 7710312401 is divisible by 11. Correct. We use the divisibility rule for 11 (Sum of digits at odd places $-$ Sum of digits at even places): Sum of odd-placed digits (from right): $1 + 4 + 1 + 0 + 7 = 13$ Sum of even-placed digits (from right): $0 + 2 + 3 + 1 + 7 = 13$ Difference: $13 - 13 = 0$ Since the difference is 0, the number is divisible by 11. Conclusion Statements (A), (B), and (D) are correct. |
