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, 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. 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. However, as we calculated, 17 is actually 2 more than the nearest multiple of 5 (which is 15), making this option incorrect. ### Summary of Key Points - The modulo operation finds the remainder of a division. - To calculate \( 17 \mod 5 \), divide 17 by 5, take the integer part (3), multiply it by 5 (15), and subtract from 17 to find the remainder (2). - The correct answer is **A. 2**. - Understanding the steps of division and remainder calculation is crucial for solving modulo problems effectively.