Consider the following statements.
I. To obtain prime number less than 121, we are to reject all the multiples of 2,3,5 and 7. If Every composite number less than 121 is divisible by a prime number less than 11.
Which of the statements given above is /are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to analyze the two statements provided and determine their correctness.

### Step 1: Analyze Statement I
**Statement I:** To obtain prime numbers less than 121, we must reject all the multiples of 2, 3, 5, and 7.

- **Understanding Prime Numbers:** A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. The prime numbers less than 121 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127.

- **Multiples of 2, 3, 5, and 7:** These numbers are not prime (except for the primes themselves) and include all even numbers and many others. However, we only need to exclude the multiples of these numbers when looking for composite numbers.

- **Conclusion for Statement I:** The statement is misleading because while we do need to consider multiples of these primes to find composite numbers, we do not need to reject all of them to find primes less than 121. Therefore, the statement is **incorrect**.

### Step 2: Analyze Statement II
**Statement II:** Every composite number less than 121 is divisible by a prime number less than 11.

- **Understanding Composite Numbers:** A composite number is a natural number greater than 1 that is not prime, meaning it has more than two distinct positive divisors.

- **Primes Less Than 11:** The prime numbers less than 11 are: 2, 3, 5, 7.

- **Checking Composite Numbers:**
- The composite numbers less than 121 include: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120.

- **Divisibility Check:** Each of these composite numbers can be divided by at least one of the primes (2, 3, 5, 7). For example:
- 4 is divisible by 2.
- 6 is divisible by 2 and 3.
- 9 is divisible by 3.
- 10 is divisible by 2 and 5.

- **Conclusion for Statement II:** Since all composite numbers less than 121 can be divided by at least one prime number less than 11, this statement is **correct**.

### Final Conclusion
- **Statement I:** Incorrect
- **Statement II:** Correct

Therefore, the answer is that only Statement II is correct.