Final answer:
To find the number of positive integers <= 490 that are not divisible by 2, 5, or 7 using Inclusion/Exclusion, we calculate the multiples of these numbers, their pairs, and then subtract these from the total of 490 to get 168.
Step-by-step explanation:
The student is asking how to find the number of positive integers less than or equal to 490 that are divisible by neither 2, nor 5, nor 7 using the Inclusion/Exclusion principle. First, we calculate the total number of multiples of 2, 5, and 7, respectively, among the first 490 numbers:
- Multiples of 2: 490 / 2 = 245
- Multiples of 5: 490 / 5 = 98
- Multiples of 7: 490 / 7 = 70
Next, we find the number of multiples of the pairs of these numbers:
- Multiples of 2 and 5 (i.e., 10): 490 / 10 = 49
- Multiples of 2 and 7 (i.e., 14): 490 / 14 = 35
- Multiples of 5 and 7 (i.e., 35): 490 / 35 = 14
And, multiples of all three numbers (i.e., 2, 5, and 7 or 70): 490 / 70 = 7
Now, we apply the Inclusion/Exclusion principle to find the number of integers divisible by neither 2, 5, nor 7:
Total = 490 - (Multiples of 2) - (Multiples of 5) - (Multiples of 7) + (Multiples of 2 and 5) + (Multiples of 2 and 7) + (Multiples of 5 and 7) - (Multiples of 2, 5, and 7)
Total = 490 - 245 - 98 - 70 + 49 + 35 + 14 - 7 = 168
Thus, there are 168 positive integers less than or equal to 490 that are divisible by neither 2, nor 5, nor 7.