7

Part of 2023 AIME

Problems(2)

Yes, there are three cases

Source:

Call a positive integer

n

extra-distinct if the remainders when

n

2, 3, 4, 5,

6

are distinct. Find the number of extra-distinct positive integers less than

1000

why so much counting maa

Source: 2023 AIME II Problem #7

Each vertex of a regular dodecagon (12-gon) is to be colored either red or blue, and thus there are

2^{12}

possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.