The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is _____.

We need to count 3-digit numbers (100-999) divisible by 2 or 3 but not by 7.

First, we count numbers divisible by 2 or 3.

Divisible by 2: $$\lfloor 999/2 \rfloor - \lfloor 99/2 \rfloor = 499 - 49 = 450$$

Divisible by 3: $$\lfloor 999/3 \rfloor - \lfloor 99/3 \rfloor = 333 - 33 = 300$$

Divisible by 6 (both 2 and 3): $$\lfloor 999/6 \rfloor - \lfloor 99/6 \rfloor = 166 - 16 = 150$$

Divisible by 2 or 3: $$450 + 300 - 150 = 600$$

Next, we subtract those also divisible by 7.

We need numbers divisible by (2 or 3) AND 7 = divisible by 14 or 21.

Divisible by 14: $$\lfloor 999/14 \rfloor - \lfloor 99/14 \rfloor = 71 - 7 = 64$$

Divisible by 21: $$\lfloor 999/21 \rfloor - \lfloor 99/21 \rfloor = 47 - 4 = 43$$

Divisible by 42 (LCM of 14 and 21): $$\lfloor 999/42 \rfloor - \lfloor 99/42 \rfloor = 23 - 2 = 21$$

Divisible by 14 or 21: $$64 + 43 - 21 = 86$$

Final answer:

Therefore, the number of 3-digit numbers is 514.