[jhshah]

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..."

[chrchang523]

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

[lupire]

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.