The correct option for the expression \( 17 \mod 5 \) is **A. 2**. ### Step-by-Step Explanation 1. **Understanding Modulo Operation**: The modulo operation, denoted by the symbol \( \mod \), finds the remainder when one number is divided by another. In this case, we are looking for the remainder when 17 is divided by 5. 2. **Performing the Division**: To find \( 17 \mod 5 \), we first divide 17 by 5: \[ 17 \div 5 = 3.4 \] However, we are interested in the integer part of this division, which is 3. This means that 5 fits into 17 a total of 3 times. 3. **Calculating the Product**: Next, we multiply the integer part of the division (which is 3) by the divisor (which is 5): \[ 3 \times 5 = 15 \] 4. **Finding the Remainder**: Now, we subtract this product from the original number (17) to find the remainder: \[ 17 - 15 = 2 \] Therefore, the remainder when 17 is divided by 5 is 2. 5. **Final Result**: Thus, we conclude that: \[ 17 \mod 5 = 2 \] ### Why Other Options Are Incorrect - **Option B (3)**: This option suggests that the remainder is 3. However, as we calculated, the integer part of the division is 3, but the remainder is what is left after subtracting the product of the divisor and the integer part from the original number, which is 2, not 3. - **Option C (4)**: This option implies that when 17 is divided by 5, the remainder is 4. This is incorrect because if the remainder were 4, it would mean that 5 could fit into 17 four times (which would be 20), exceeding 17. - **Option D (1)**: This option suggests that the remainder is 1. However, as shown in our calculations, the remainder is 2, not 1. If the remainder were 1, it would imply that 17 is just 1 more than a multiple of 5, which is not the case here. ### Common Pitfalls - **Confusing Quotient with Remainder**: A common mistake is to confuse the quotient (the integer part of the division) with the remainder. Remember, the remainder is what is left after subtracting the largest multiple of the divisor that fits into the original number. - **Misunderstanding Modulo**: Some students may not fully grasp the concept of modulo and may think it simply involves division without considering the remainder. ### Revision Summary - The modulo operation finds the remainder of a division. - To calculate \( a \mod b \), divide \( a \) by \( b \), take the integer part of the quotient, multiply it by \( b \), and subtract from \( a \). - For \( 17 \mod 5 \), the remainder is 2. - Always ensure to differentiate between the quotient and the remainder in division problems.