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.

(B) Every composite number less than 121 is divisible by a prime number less than 11.

(C) 173 is not a prime number.

(D) 7710312401 is divisible by 11.

Which of the statement(s) given above is/are correct?

Choose the correct answer from the options given below:

To find prime numbers up to any number using the Sieve of Eratosthenes, reject multiples of primes up to the square root of that number.

$\sqrt{121} = 11$

Primes less than 11 are: 2, 3, 5, 7

The statement says to reject multiples of 2, 3, 5, and 7.

Statement (A) is correct.

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Any composite number must have at least one prime factor that is less than or equal to its square root. If a composite number $n = a \times b$, at least one factor must be $\leq \sqrt{n}$.

For numbers less than 121:

$\sqrt{121} = 11$

Every composite number less than 121 must have a prime factor less than 11.

Statement (B) is correct.

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To check if 173 is prime, test divisibility by all primes up to $\sqrt{173} \approx 13.15$

Primes to check: 2, 3, 5, 7, 11, 13

$173$ is odd, not divisible by 2

Sum of digits $= 1+7+3 = 11$, not divisible by 3

Doesn't end in 0 or 5, not divisible by 5

$173 \div 7 = 24.71...$, not divisible by 7

$173 \div 11 = 15.72...$, not divisible by 11

$173 \div 13 = 13.31...$, not divisible by 13

Since 173 is not divisible by any prime up to $\sqrt{173}$, the number 173 is prime.

Statement (C) is incorrect.

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Divisibility rule for 11: Sum the digits at odd positions (1st, 3rd, 5th...) and sum the digits at even positions (2nd, 4th, 6th...). If the difference is 0 or divisible by 11, then the number is divisible by 11.

Number: 7710312401

Odd positions (1,3,5,7,9): $7 + 1 + 3 + 2 + 0 = 13$

Even positions (2,4,6,8,10): $7 + 0 + 1 + 4 + 1 = 13$

Difference: $13 - 13 = 0$

Since the difference is 0, the number is divisible by 11.

Statement (D) is correct.

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Statements (A), (B), and (D) are correct.