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 the remainder is 4. For this to be true, 5 would have to fit into 17 four times, which would give a product of 20 (5 times 4). Since 20 is greater than 17, this option is incorrect. - **Option D (1)**: This option suggests that the remainder is 1. If we were to have a remainder of 1, it would mean that 17 is 1 more than a multiple of 5. The closest multiple of 5 less than 17 is 15, and the difference is indeed 2, not 1. ### Common Pitfalls - **Confusing the Quotient with the Remainder**: A common mistake is to take the integer part of the division (the quotient) as the answer instead of finding the remainder. - **Misunderstanding the Modulo Definition**: Remember that \( a \mod b \) is defined as the remainder of the division of \( a \) by \( b \), not the result of the division itself. ### Revision Summary - The modulo operation finds the remainder of a division. - To calculate \( a \mod b \), divide \( a \) by \( b \), take the integer part, multiply back, and subtract from \( a \). - For \( 17 \mod 5 \), the remainder is 2. - Always ensure to differentiate between the quotient and the remainder in modulo calculations.