2. How many natural numbers less than 1000 are not divisible by either 7 or 11?
2. Since $999 : 7 = 142$ with a remainder of 5, we conclude that there are 142 numbers less than 1000 that are divisible by 7. $\quad 2$ POINTS Since $999 : 11 = 90$ with a remainder of 9, we conclude that there are 90 numbers less than 1000 that are divisible by 11. 2 POINTS Therefore, we need to subtract from 999 the number of numbers that are divisible by 7 and 11. However, among the numbers divisible by 7, there are those that are divisible by 11, and vice versa, among the numbers divisible by 11, there are those that are divisible by 7. In both cases, these are numbers that are divisible by $11 \cdot 7 = 77$. $\quad 1$ POINT Since $999 : 77 = 12$ with a remainder of 75, we conclude that there are 12 numbers less than 1000 that are divisible by 77. 2 POINTS Therefore, the number of the desired numbers is equal to $999 - 142 - 90 + 12 = 779$, because we subtracted the numbers divisible by 77 twice, so we had to add them once.