To solve the problem \( 25 \mod 7 \), we need to understand what the modulus operation means. The expression \( a \mod b \) gives us the remainder when \( a \) is divided by \( b \). In this case, we are looking for the remainder when 25 is divided by 7. ### Step-by-Step Explanation: 1. **Division**: First, we divide 25 by 7. - When we perform the division, we find how many times 7 can fit into 25 without exceeding it. - \( 25 \div 7 = 3.5714 \) (approximately), but we are interested in the integer part of this division, which is 3. 2. **Multiplication**: Next, we multiply the integer part of the division (which is 3) by 7 to find out how much of 25 is accounted for by multiples of 7. - \( 3 \times 7 = 21 \) 3. **Subtraction**: Now, we subtract this product from 25 to find the remainder. - \( 25 - 21 = 4 \) 4. **Conclusion**: The remainder, which is the result of \( 25 \mod 7 \), is 4. Thus, the correct answer is **B. 4**. ### Why Other Options Are Incorrect: - **Option A: 3**: This option suggests that the remainder is 3. However, as we calculated, the remainder after dividing 25 by 7 is actually 4, not 3. Therefore, this option is incorrect. - **Option C: 5**: This option implies that after dividing 25 by 7, the remainder is 5. However, since 21 (which is \( 3 \times 7 \)) is the largest multiple of 7 less than 25, the remainder cannot be 5. Thus, this option is also incorrect. - **Option D: 6**: This option suggests that the remainder is 6. If we had a remainder of 6, that would mean that \( 25 - 6 = 19 \) is a multiple of 7, which it is not (since \( 19 \div 7 \) does not yield an integer). Therefore, this option is incorrect as well. ### Common Pitfalls: - **Misunderstanding Modulus**: A common mistake is to confuse the modulus operation with simple division. Remember, modulus gives the remainder, not the quotient. - **Rounding Errors**: When performing the division, ensure you only take the integer part (the floor of the division) and not the decimal. - **Forgetting to Subtract**: After finding the largest multiple of the divisor, always remember to subtract it from the original number to find the remainder. ### Revision Summary: - The modulus operation \( a \mod b \) gives the remainder of \( a \) divided by \( b \). - To find \( 25 \mod 7 \), divide 25 by 7, take the integer part (3), multiply by 7 (21), and subtract from 25 to get the remainder (4). - The correct answer is **B. 4**. - Always remember to check your calculations and understand the difference between quotient and remainder.