1. The problem is to find the number of integers from 1 to 1000 that are not divisible by 2, 3, or 5. 2. We use the principle of inclusion-exclusion to solve this. The formula for the number of elements not having any of the properties $S_1, S_2, \ldots, S_k$ is: $$N - \sum N_x + \sum N_{x,y} - \sum N_{x,y,z} + \cdots$$ where $N$ is the total number of objects, $N_x$ is the number of objects with property $S_x$, $N_{x,y}$ is the number with both $S_x$ and $S_y$, and so on. 3. Here, $N=1000$ (numbers 1 to 1000). 4. Define the sets: - $S_1$: divisible by 2, so $N_1 = \lfloor \frac{1000}{2} \rfloor = 500$ - $S_2$: divisible by 3, so $N_2 = \lfloor \frac{1000}{3} \rfloor = 333$ - $S_3$: divisible by 5, so $N_3 = \lfloor \frac{1000}{5} \rfloor = 200$ 5. Intersections: - $N_{1,2} = \lfloor \frac{1000}{2 \times 3} \rfloor = 166$ - $N_{1,3} = \lfloor \frac{1000}{2 \times 5} \rfloor = 100$ - $N_{2,3} = \lfloor \frac{1000}{3 \times 5} \rfloor = 66$ - $N_{1,2,3} = \lfloor \frac{1000}{2 \times 3 \times 5} \rfloor = 33$ 6. Applying inclusion-exclusion: $$\text{Count} = N - (N_1 + N_2 + N_3) + (N_{1,2} + N_{1,3} + N_{2,3}) - N_{1,2,3}$$ $$= 1000 - (500 + 333 + 200) + (166 + 100 + 66) - 33$$ 7. Calculate step-by-step: - Sum singles: $500 + 333 + 200 = 1033$ - Sum doubles: $166 + 100 + 66 = 332$ - Final count: $$1000 - 1033 + 332 - 33 = (1000 - 1033) + 332 - 33 = -33 + 332 - 33 = 266$$ **Final answer:** There are $\boxed{266}$ integers between 1 and 1000 that are not divisible by 2, 3, or 5.