Agree, and to add a bit more context:

* The contest has 15 problems and a time limit of 3 hours, giving only an average of 12 minutes per problem.

* The average score is 4.83/15, from the official statistics [1].

* The statistics noted that only 5.17% of test takers answered this problem correctly, making it the second-hardest problem out of the 15 on the exam.

[1] "AIME 02/08/2022" https://amc-reg.maa.org/reports/generalreports.aspx

What's especially fascinating is that the core of the problem is a generalization of the totient computation, so understanding the inclusion-exclusion construction of totient is very helpful to the problem, while simply memorizing the formula is a misdirection.

OP missing this point shows that he really was doing this for all the wrong reasons. He should have done FIRST or science bowl instead.

Completely agree, I also liked the problem and thought it was conceptual as far as these things go.

Asking for N mod 1000 was another cute twist that was meant to get you thinking about the divisibility properties of the totient function - "hmm, so (p-1) always divides \phi(n) for all prime factors p of n, how convenient..."

The mod 1000 is actually a consequence of the test format: all answers are integers in [0, 999], you fill in 3 digit-bubbles.

That's a cute coincidence that 1000 divides (11-1)(101-1), but I doubt it was intentional. The test writers didn't have much choice unless they chose a different base than 5+5.