Ask AI: How many postive integers less than 100 is not a factor of 2,3,5?

To determine how many positive integers less than 100 are not factors of 2, 3, or 5, we can use the principle of inclusion-exclusion. First, let's find the number of integers less than 100 that are divisible by each of these numbers. 1. **Multiples of 2:** There are \(\left\lfloor \frac{99}{2} \right\rfloor = 49\) multiples of 2 less than 100 (because \(\left\lfloor 49.5 \right\rfloor = 49\)). 2. **Multiples of 3:** There are \(\left\lfloor \frac{99}{3} \right\rfloor = 33\) multiples of 3 less than 100. 3. **Multiples of 5:** There are \(\left\lfloor \frac{99}{5} \right\rfloor = 19\) multiples of 5 less than 100. Next, we apply the principle of inclusion-exclusion to find the number of integers that are multiples of 2, 3, or 5. 4. **Multiples of 2 and 3 (i.e., multiples of 6):** There are \(\left\lfloor \frac{99}{6} \right\rfloor = 16\) multiples of 6 less than 100. 5. **Multiples of 2 and 5 (i.e., multiples of 10):** There are \(\left\lfloor \frac{99}{10} \right\rfloor = 9\) multiples of 10 less than 100. 6. **Multiples of 3 and 5 (i.e., multiples of 15):** There are \(\left\lfloor \frac{99}{15} \right\rfloor = 6\) multiples of 15 less than 100. 7. **Multiples of 2, 3, and 5 (i.e., multiples of 30):** There are \(\left\lfloor \frac{99}{30} \right\rfloor = 3\) multiples of 30 less than 100. Using inclusion-exclusion: Number = 49 + 33 + 19 - 16 - 9 - 6 + 3 = 73 So, there are 73 integers less than 100 that are multiples of 2, 3, or 5. Finally, there are \(99 - 73 = 26\) integers less than 100 that are **not** multiples of 2, 3, or 5. Thus, the number of positive integers less than 100 that are not a factor of 2, 3, or 5 is 26.

A positive integer less than 100 that is not a factor of 2, 3, or 5 can only be a prime number greater than 5. Here are those primes: 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. So there are 22 positive integers less than 100 that are not factors of 2, 3 or 5.