1. The problem asks to find which number among 373, 450, 555, 128, and 261 has the greatest remainder when divided by 30. 2. The remainder when a number $n$ is divided by $d$ is found using the formula: $$\text{remainder} = n - d \times \left\lfloor \frac{n}{d} \right\rfloor$$ where $\left\lfloor x \right\rfloor$ is the floor function (greatest integer less than or equal to $x$). 3. Calculate the remainder for each number: - For 373: $$\left\lfloor \frac{373}{30} \right\rfloor = 12$$ $$\text{remainder} = 373 - 30 \times 12 = 373 - 360 = 13$$ - For 450: $$\left\lfloor \frac{450}{30} \right\rfloor = 15$$ $$\text{remainder} = 450 - 30 \times 15 = 450 - 450 = 0$$ - For 555: $$\left\lfloor \frac{555}{30} \right\rfloor = 18$$ $$\text{remainder} = 555 - 30 \times 18 = 555 - 540 = 15$$ - For 128: $$\left\lfloor \frac{128}{30} \right\rfloor = 4$$ $$\text{remainder} = 128 - 30 \times 4 = 128 - 120 = 8$$ - For 261: $$\left\lfloor \frac{261}{30} \right\rfloor = 8$$ $$\text{remainder} = 261 - 30 \times 8 = 261 - 240 = 21$$ 4. Compare the remainders: 13, 0, 15, 8, 21. 5. The greatest remainder is 21, which corresponds to the number 261. **Final answer:** The number 261 has the greatest remainder when divided by 30, which is 21.