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: \[ 17 - 15 = 2 \] - This means that when we divide 17 by 5, we have a remainder of 2. 3. **Conclusion**: - Therefore, \( 17 \mod 5 = 2 \). ### Why the Correct Option 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. ### Why the Other Options are Incorrect: - **B. 1**: This option suggests that the remainder is 1. However, as we calculated, the remainder when dividing 17 by 5 is 2, not 1. Thus, this option is incorrect. - **C. 3**: This option implies that the remainder is 3. If we had a remainder of 3, it would mean that \( 17 - 5 \times 3 = 2 \), which is not the case. The correct calculation shows that the remainder is 2, making this option incorrect. - **D. 4**: This option suggests that the remainder is 4. If that were true, it would mean that \( 17 - 5 \times 3 = 4 \), which is not correct. The correct calculation shows that the remainder is 2, so this option is also incorrect. ### 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. - **Rounding Errors**: When dividing, ensure you are looking for the largest whole number that fits into the dividend without exceeding it. - **Forgetting to Subtract**: Always remember to subtract the product of the divisor and the quotient from the original number to find the remainder. ### Revision Summary: - The modulus operation finds the remainder of a division. - To calculate \( 17 \mod 5 \), divide 17 by 5 and find the remainder. - The correct answer is A (2), as \( 17 - 15 = 2 \). - Other options (B, C, D) are incorrect because they do not represent the actual remainder from the division. By understanding these concepts, you can confidently tackle similar modulus problems in the future!