Math Gold Medalist

Lor

2023 AIME I 

Problem 7

Call a positive integer $n$ extra-distinct if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$.

Casework

Priority: The Most Limited Case

Chinese Remainder Theorem

Change Multiple Mods to 1 Mod

Writing All Cases

Find impossible remainders mod 4, 6, 10, 12, 20.

Then write 1 to 60 and cross out numbers that have impossible remainders.

Then few numbers will be left and check them to satisfy assumption of the problem.

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   Solution