To solve the problem \( 17 \mod 5 \), we need to understand what the modulus operation means. The modulus operation finds the remainder when one number is divided by another. In this case, we are dividing 17 by 5 and looking for the remainder. ### Step-by-Step Explanation: 1. **Division**: First, we divide 17 by 5. - When we perform the division, we can see how many times 5 fits into 17. - \( 5 \times 3 = 15 \) is the largest multiple of 5 that is less than or equal to 17. - If we try \( 5 \times 4 = 20 \), it exceeds 17, so we stick with 3. 2. **Finding the Remainder**: - Now, we subtract the largest multiple of 5 that fits into 17 from 17 itself: \[ 17 - 15 = 2 \] - Therefore, the remainder when 17 is divided by 5 is 2. 3. **Conclusion**: - Thus, \( 17 \mod 5 = 2 \). ### Why the Correct Answer is A (2): The correct answer is A (2) because we have calculated the remainder of the division of 17 by 5, which is indeed 2. ### Explanation of Other Options: - **Option B (3)**: This option is incorrect because it suggests that the remainder when dividing 17 by 5 is 3. However, as we calculated, the actual remainder is 2, not 3. - **Option C (4)**: This option is also incorrect. If the remainder were 4, it would imply that \( 17 - 5 \times 3 = 4 \), which is not true since \( 17 - 15 = 2 \). - **Option D (1)**: This option is incorrect as well. A remainder of 1 would mean that \( 17 - 5 \times 3 = 1 \), which again does not hold true since we found the remainder to be 2. ### Common Pitfalls: - **Misunderstanding Modulus**: A common mistake is to confuse the modulus operation with simple division. Remember, modulus gives you the remainder, not the quotient. - **Forgetting to Subtract**: Some students might forget to subtract the largest multiple of the divisor from the dividend, leading to incorrect remainders. ### Revision Summary: - The modulus operation finds the remainder of a division. - To calculate \( a \mod b \), divide \( a \) by \( b \) and find the remainder. - For \( 17 \mod 5 \), the calculation shows that the remainder is 2. - The correct answer is A (2), while options B (3), C (4), and D (1) are incorrect. By understanding these concepts, you can confidently tackle similar modulus problems in the future!