Question

How many positive integers less or equal 100 are not multiples of 2 or 3 or 5?

Answer 1

To find the number of positive integers less than or equal to 100 that are not multiples of 2, 3, or 5, count the multiples of each integer, adjust for overlaps, and subtract from 100, yielding 26 such numbers.

Step-by-step explanation:

The question asks us to find the number of positive integers less than or equal to 100 that are not multiples of 2, 3, or 5. To solve this, we will find the total number of multiples of 2, 3, and 5 up to 100 and then subtract this from the total count of numbers from 1 to 100.

Steps to Determine Non-Multiples:

1. Count the multiples of 2, 3, and 5 separately.
2. Subtract the overlapping counts (numbers which are multiples of more than one of 2, 3, or 5).
3. Add back the count for numbers that are multiples of the least common multiple (LCM) of 2, 3, and 5, which is 30, because we subtracted them twice.
4. Subtract the count of all multiples (including overlaps) from the total numbers from 1 to 100.

Now, let's perform the calculations:

Multiples of 2 (up to 100): 100/2 = 50
Multiples of 3 (up to 100): 100/3 = 33 (discard the remainder)
Multiples of 5 (up to 100): 100/5 = 20
Multiples of 2 and 3 (LCM is 6): 100/6 = 16 (discard the remainder)
Multiples of 2 and 5 (LCM is 10): 100/10 = 10
Multiples of 3 and 5 (LCM is 15): 100/15 = 6 (discard the remainder)
Multiples of 2, 3, and 5 (LCM is 30): 100/30 = 3 (discard the remainder)

Now, subtracting the overlaps:

Total number of multiples: 50 + 33 + 20 = 103
Subtracting overlaps (two at a time): 103 - (16 + 10 + 6) = 71
Adding overlaps (all three multiples): 71 + 3 = 74
Total numbers from 1 to 100: 100
Non-multiples of 2, 3, or 5: 100 - 74 = 26

Therefore, there are 26 positive integers less than or equal to 100 that are not multiples of 2, 3, or 5.