Consider the following statements :
I. 7710312401, is divisible by 11.
II. 173 is a prime number.
Which of the statements given above is/are correct?

Question

Consider the following statements : 
I. 7710312401, is divisible by 11. 
II. 173 is a prime number. 
Which of the statements given above is/are correct?

Accepted Answer

To determine why Option 3 is the correct answer, we need to evaluate both statements individually and confirm their validity.

**Statement I: 7710312401 is divisible by 11.**

To check if a number is divisible by 11, we use the divisibility rule for 11, which states that the difference between the sum of the digits in odd positions and the sum of the digits in even positions should be either 0 or a multiple of 11.

For the number 7710312401:
- Odd-positioned digits: 7, 1, 3, 2, 0 (from left to right)
- Even-positioned digits: 7, 0, 1, 4, 1

Sum of odd-positioned digits = 7 + 1 + 3 + 2 + 0 = 13
Sum of even-positioned digits = 7 + 0 + 1 + 4 + 1 = 13

Difference = 13 - 13 = 0

Since the difference is 0, which is a multiple of 11, the number 7710312401 is divisible by 11. Therefore, Statement I is correct.

**Statement II: 173 is a prime number.**

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. To determine if 173 is a prime number, we check for divisibility by prime numbers less than the square root of 173 (approximately 13.15). The prime numbers to check are 2, 3, 5, 7, 11, and 13.

- 173 is odd, so it is not divisible by 2.
- The sum of the digits of 173 is 1 + 7 + 3 = 11, which is not divisible by 3.
- 173 does not end in 0 or 5, so it is not divisible by 5.
- Dividing 173 by 7 gives approximately 24.71, which is not an integer.
- Dividing 173 by 11 gives approximately 15.73, which is not an integer.
- Dividing 173 by 13 gives approximately 13.31, which is not an integer.

Since 173 is not divisible by any prime number up to 13, it is a prime number. Therefore, Statement II is correct.

**Conclusion:**

Both Statement I and Statement II are correct. Therefore, the correct answer is Option 3: Both I and II.

**Explanation of Incorrect Options:**

1. **Option 1 (I only):** This option is incorrect because it ignores the correctness of Statement II. As shown, Statement II is also correct.

2. **Option 2 (II only):** This option is incorrect because it ignores the correctness of Statement I. As demonstrated, Statement I is also correct.

4. **Option 4 (Neither I nor II):** This option is incorrect because both statements are correct, as explained above.

Thus, the correct answer is Option 3: Both I and II.