Janaína bought a different deck: there are 180 cards numbered from 1 to 180, with the even cards being red and the odd cards being blue.

a) How many cards are multiples of 7 and red?

b) How many cards are multiples of 7 or red?

c) Janaína randomly chose 80 cards, all, coincidentally, black. What is the minimum number of cards that are multiples of 7?
Solution

a) Since the red cards are even, the red cards that are multiples of 7 must also be multiples of 2, that is, they must be multiples of 14. The first multiple of 14 is $14=1 \cdot 14$, while the last is $168=12 \cdot 14$, which means there are 12 red cards that are multiples of 7.

b) The first odd multiple of 7 is $7=1 \cdot 7$ and the last is $175=25 \cdot 7$, which means there are 13 odd multiples of 7. Therefore, the number of red cards or multiples of 7 is $90+13=103$.
c) Since there are 13 blue cards that are multiples of 7, the minimum number of them, among the 80 blue cards, is 3, because the maximum number of blue multiples of 7 not chosen is 10.