In the Year 2525

Here's a little puzzle I wrote:

The song wonders, “In the year 2525, if man is still alive . . . ”

In what year must man be alive to hear the cicadas and see Halley’s comet in the same year?

Background:
The cicadas last appeared in 2004 and return every 17 years.
Halley's Comet last appeared in 1986 and returns every 76 years.

Wow. That'll get your mind going. I thought the answer was never since 76 and 17 are relatively prime, and the difference in 1986 and 2004 is 18, but of course you can't say never without a proof. When trying to prove it, I came up with an answer. The answer is the year 3126. This is 15 Haley's Comets and 66 locust seasons away.

n = 2004 mod 17 and n = 1986 mod 76
n - 1986 = 76m
n - 2004 = 76m - 18
17 | n - 2004 => 17 | 76m - 18
76m = 19 mod 17
m = 15

I have the solution here in a more verbose and didactic style.

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Please note - this is on my old web site - entropy is taking over.