Capitu has a hundred cards numbered from 1 to 100. All cards have one yellow side and the other red, and the number of each card is written on both sides. The cards were placed on a table, all with the red side facing up. Capitu flipped all the cards with even numbers and then all the cards with numbers that are multiples of 3, placing them with the yellow side facing up. How many cards remained with the red side facing up?
Solution 1: First, Capitu turned over the 50 even-numbered cards. After that, the 50 even-numbered cards were face up with the yellow side, and the 50 odd-numbered cards were face up with the red side. When she then turned over the multiples of 3, she only turned over the odd multiples of 3, which are 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, and 99. Therefore, there are 17 odd multiples of 3, and Capitu turned the yellow side up on $50+17=67$ cards. Thus, $100-67=33$ cards remained with the red side up.

Note: In this solution, to determine the number of odd multiples of 3 less than 100, it was sufficient to list these multiples and count them. However, if Capitu had 1000 cards (or more), this method would be quite laborious. In such a case, we can proceed in a more general way. We notice that the odd multiples of 3 from 1 to 1000 form an arithmetic progression, with the first term $a_{1}=3$, common difference $r=6$, and the last term $a_{n}=999$. To determine $n$, we use the formula $a_{n}=a_{1}+(n-1) r$, which in this case is $999=3+(n-1) \times 6$. Thus, $n=167$, meaning there are 167 odd multiples of 3 less than 1000.

Solution 2: First, Capitu turned over the 50 even-numbered cards. After that, the 50 even-numbered cards were face up with the yellow side, and the 50 odd-numbered cards were face up with the red side. When she then turned over the multiples of 3, Capitu proceeded as follows.

- Among the even-numbered cards, she turned over those that were also multiples of 3. A number that is a multiple of 2 and 3 is also a multiple of 6. Since $100=16 \times 6+4$, we conclude that Capitu turned over 16 cards among the even-numbered cards. These cards returned to having the red side up, leaving 34 with the yellow side up.
- Among the odd-numbered cards, since $100=33 \times 3+1$, the total number of cards (even and odd) that are multiples of 3 is 33. As we saw above, 16 of these are even, so 17 are odd. Thus, Capitu turned over 17 odd-numbered cards, and these cards now had the yellow side up, while the other 33 continued to have the red side up.