Question 58

1. Sample Space:
   Integers from 1 to 100 inclusive, so $N = 100$.

2. Counting Multiples:

   • Divisible by 2: $|A_2| = \lfloor 100 / 2 \rfloor = 50$.
   • Divisible by 3: $|A_3| = \lfloor 100 / 3 \rfloor = 33$.
   • Divisible by 5: $|A_5| = \lfloor 100 / 5 \rfloor = 20$.

3. Pairwise and Three-way Intersections:

   • Divisible by $\text{lcm}(2, 3) = 6$: $|A_2 \cap A_3| = \lfloor 100 / 6 \rfloor = 16$.
   • Divisible by $\text{lcm}(2, 5) = 10$: $|A_2 \cap A_5| = \lfloor 100 / 10 \rfloor = 10$.
   • Divisible by $\text{lcm}(3, 5) = 15$: $|A_3 \cap A_5| = \lfloor 100 / 15 \rfloor = 6$.
   • Divisible by $\text{lcm}(2, 3, 5) = 30$: $|A_2 \cap A_3 \cap A_5| = \lfloor 100 / 30 \rfloor = 3$.

4. Inclusion-Exclusion Calculation:

   $$|A_2 \cup A_3 \cup A_5| = 50 + 33 + 20 - (16 + 10 + 6) + 3 = 103 - 32 + 3 = 74.$$

5. Count and Probability of NOT Divisible:

   • Integers not divisible by 2, 3, or 5:
     $$100 - 74 = 26.$$
   • Probability:
     $$P = \frac{26}{100} = 0.26.$$

This lies squarely in the official key range $[0.259, 0.261]$.