1. **Problem 1: Divide the numbers using long division.** (1) Divide 2812 by 36. (2) Divide 1886 by 46. (3) Divide 2915 by 53. (4) Divide 2144 by 67. --- 2. **Problem 2: Probability of drawing a pyramid, a cylinder, and a hexagon from given figures.** Pudge has 5 solid figures and 5 plane figures. He will draw 3 figures. Find the probability that he draws a pyramid, a cylinder, and a hexagon. --- 3. **Problem 3: Ratio of solid figures to plane figures drawn.** Pudge drew 2 solid figures and 5 plane figures. Find the ratio of solid figures to plane figures. --- ### Step 1: Divide 2812 by 36 $$36 \times 78 = 2808$$ Remainder: $$2812 - 2808 = 4$$ So, $$2812 \div 36 = 78 \text{ remainder } 4$$ ### Step 2: Divide 1886 by 46 $$46 \times 41 = 1886$$ Remainder: $$1886 - 1886 = 0$$ So, $$1886 \div 46 = 41$$ ### Step 3: Divide 2915 by 53 $$53 \times 55 = 2915$$ Remainder: $$2915 - 2915 = 0$$ So, $$2915 \div 53 = 55$$ ### Step 4: Divide 2144 by 67 $$67 \times 32 = 2144$$ Remainder: $$2144 - 2144 = 0$$ So, $$2144 \div 67 = 32$$ --- ### Step 5: Probability of drawing a pyramid, a cylinder, and a hexagon - Total figures: $$5 \text{ solid} + 5 \text{ plane} = 10$$ - Figures to draw: 3 - The figures to draw are pyramid (solid), cylinder (solid), and hexagon (plane). Number of ways to choose these 3 specific figures is 1 (since they are specific). Total number of ways to choose any 3 figures from 10 is: $$\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$$ Therefore, the probability is: $$\frac{1}{120}$$ --- ### Step 6: Ratio of solid figures to plane figures drawn Pudge drew 2 solid figures and 5 plane figures. Ratio solid : plane = $$2 : 5$$ --- **Final answers:** 1. $$2812 \div 36 = 78 \text{ remainder } 4$$ 2. $$1886 \div 46 = 41$$ 3. $$2915 \div 53 = 55$$ 4. $$2144 \div 67 = 32$$ 5. Probability of drawing pyramid, cylinder, and hexagon = $$\frac{1}{120}$$ 6. Ratio of solid to plane figures drawn = $$2 : 5$$