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:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
A composite number \(n\) always has a prime factor not greater than \(\sqrt{n}\): if both factors were larger than \(\sqrt{n}\), their product would be larger than \(n\). This decides (A), (B) and (C).
(A) To obtain prime numbers less than 121, we have to reject all the multiples of 2, 3, 5 and 7. Correct. \(\sqrt{121}=11\), so every composite number below 121 has a prime factor below 11, that is, 2, 3, 5 or 7. Striking out their multiples (the sieve of Eratosthenes) leaves only the primes.
(B) Every composite number less than 121 is divisible by a prime number less than 11. Correct. This is the same fact: the smallest composite number with no prime factor below 11 is \(11\times11=121\) itself.
(C) 173 is not a prime number. Incorrect. \(\sqrt{173}\) is a little over 13, and 173 is not divisible by 2, 3, 5, 7, 11 or 13, so 173 is a prime number.
(D) 7710312401 is divisible by 11. Correct. A number is divisible by 11 when the sums of its alternate digits differ by 0 or a multiple of 11.
Digits in odd places from the right: \(1+4+1+0+7=13\)
Digits in even places from the right: \(0+2+3+1+7=13\)
Difference \(=13-13=0\), so the number is divisible by 11