1. **State the problem:** We need to find how many whole numbers from 1 to 100 are multiples of 5 or 7. 2. **Formula and rules:** To find the count of numbers divisible by 5 or 7, we use the principle of inclusion-exclusion: $$\text{Count}(5 \cup 7) = \text{Count}(5) + \text{Count}(7) - \text{Count}(5 \cap 7)$$ where: - $\text{Count}(5)$ is the number of multiples of 5, - $\text{Count}(7)$ is the number of multiples of 7, - $\text{Count}(5 \cap 7)$ is the number of multiples of both 5 and 7 (i.e., multiples of 35). 3. **Calculate multiples of 5:** The multiples of 5 between 1 and 100 are $5, 10, 15, ..., 100$. Number of multiples of 5 is: $$\left\lfloor \frac{100}{5} \right\rfloor = 20$$ 4. **Calculate multiples of 7:** The multiples of 7 between 1 and 100 are $7, 14, 21, ..., 98$. Number of multiples of 7 is: $$\left\lfloor \frac{100}{7} \right\rfloor = 14$$ 5. **Calculate multiples of both 5 and 7 (i.e., 35):** The multiples of 35 between 1 and 100 are $35, 70$. Number of multiples of 35 is: $$\left\lfloor \frac{100}{35} \right\rfloor = 2$$ 6. **Apply inclusion-exclusion principle:** $$\text{Count}(5 \cup 7) = 20 + 14 - 2 = 32$$ **Final answer:** There are **32** whole numbers from 1 to 100 that are multiples of 5 or 7.