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. **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 if the remainder were 4, it would mean that 5 could fit into 17 four times, which is not possible since \( 5 \times 4 = 20 \), which exceeds 17. - **Option D (1)**: This option suggests that the remainder is 1. However, as shown in our calculations, the remainder is 2. If the remainder were 1, it would imply that \( 17 - 5 \times 3 = 1 \), which is incorrect since \( 17 - 15 = 2 \). ### Common Pitfalls - **Confusing Division with Remainder**: Students often confuse the quotient (the result of the division) with the remainder. Remember, the modulo operation specifically asks for the remainder. - **Misunderstanding the Modulo Concept**: It’s important to remember that the modulo operation is about what is left over after division, not how many times the divisor fits into the dividend. ### 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 your calculations to avoid confusing the quotient with the remainder.