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 perform the division of 17 by 5: \[ 17 \div 5 = 3.4 \] However, for the modulo operation, we are interested in the integer part of the 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. **Conclusion**: 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. The correct remainder 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 4 is greater than the divisor (5) and cannot be a valid remainder in this context. - **Option D (1)**: This option suggests that the remainder is 1. However, as shown in our calculations, the remainder is 2, not 1. ### Common Pitfalls - **Confusing Quotient with Remainder**: A common mistake is to confuse the quotient (the result of the division) with the remainder. Remember, the modulo operation specifically asks for the remainder. - **Remainders Greater than the Divisor**: Always ensure that the remainder is less than the divisor. In this case, since 5 is the divisor, the remainder must be less than 5. ### 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 check that the remainder is less than the divisor to avoid errors.