How many integers from 1 to 1000 are divisible by 30 but not by 16?

There are 29 integers from 1 to 1000 that are divisible by 30 but not by 16.

Here's how we arrive at that answer:

1. Find the multiples of 30 within the range: We need to determine how many multiples of 30 exist between 1 and 1000. We can do this by dividing 1000 by 30: 1000 / 30 = 33.33. This means there are 33 multiples of 30 between 1 and 1000.

2. Identify multiples of 30 that are also multiples of 16: A number divisible by both 30 and 16 must be divisible by the least common multiple (LCM) of 30 and 16.

   • Prime factorization of 30: 2 x 3 x 5
   • Prime factorization of 16: 2⁴
   • LCM(30, 16) = 2⁴ x 3 x 5 = 16 x 15 = 240

3. Find the multiples of 240 within the range: We now need to find how many multiples of 240 exist between 1 and 1000. We can do this by dividing 1000 by 240: 1000 / 240 = 4.166. This means there are 4 multiples of 240 between 1 and 1000 (240, 480, 720, 960).

4. Subtract the multiples of both 30 and 16 from the multiples of 30: We have 33 multiples of 30, and 4 of those are also multiples of 16. Therefore, we subtract 4 from 33: 33 - 4 = 29.

Therefore, there are 29 integers from 1 to 1000 that are divisible by 30 but not by 16.