1. The problem is to find the remainder when -41 is divided by 9, i.e., compute $-41 \bmod 9$. 2. Recall that the modulo operation finds the remainder after division, and the result is always between 0 and one less than the divisor (here between 0 and 8). 3. First, divide -41 by 9: $$-41 \div 9 = -4.555\ldots$$ 4. The integer quotient is the floor of this division, which is $-5$ because floor rounds down to the next lower integer. 5. Multiply the quotient by the divisor: $$-5 \times 9 = -45$$ 6. Subtract this from the original number to find the remainder: $$-41 - (-45) = -41 + 45 = 4$$ 7. Therefore, $$-41 \bmod 9 = 4$$. The remainder is 4.