The number of positive integers less than or equal to 100 that are divisible by none of 2, 3, and 5 is 26. This can be determined using the Principle of Inclusion-Exclusion.
Here's a breakdown:
Let A be the set of integers divisible by 2, B be the set of integers divisible by 3, and C be the set of integers divisible by 5, all within the range of 1 to 100.
- $|A| = \lfloor \frac{100}{2} \rfloor = 50$
- $|B| = \lfloor \frac{100}{3} \rfloor = 33$
- $|C| = \lfloor \frac{100}{5} \rfloor = 20$
- $|A \cap B| = \lfloor \frac{100}{6} \rfloor = 16$ (divisible by both 2 and 3, i.e., divisible by 6)
- $|A \cap C| = \lfloor \frac{100}{10} \rfloor = 10$ (divisible by both 2 and 5, i.e., divisible by 10)
- $|B \cap C| = \lfloor \frac{100}{15} \rfloor = 6$ (divisible by both 3 and 5, i.e., divisible by 15)
- $|A \cap B \cap C| = \lfloor \frac{100}{30} \rfloor = 3$ (divisible by 2, 3, and 5, i.e., divisible by 30)
Using the Principle of Inclusion-Exclusion:
$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$
$|A \cup B \cup C| = 50 + 33 + 20 - 16 - 10 - 6 + 3 = 74$
This represents the number of integers divisible by 2, 3, or 5. To find the number of integers not divisible by 2, 3, or 5, we subtract this from the total number of integers (100):
$100 - |A \cup B \cup C| = 100 - 74 = 26$
Therefore, 26 positive integers less than or equal to 100 are not divisible by 2, 3, or 5.
Reference Confirmation:
The provided reference indicates that from 91 to 100, only 91 and 97 are not co-prime to 2, 3, or 5 and that the correct option will be 26. The statement should be: from 91 to 100, only 91, 92, 93, 94, 95, 96, 98, 99, and 100 are co-prime to 2, 3 or 5. Hence correct option will be 26. This confirms the final answer.
