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A math problem, proposed to me by my friend Nick Gunther.

There's a nice solution using some college-level math,
but there's an even nicer elementary argument, accessible to anyone who
did pretty well in Algebra 2.

Look at the number t = 6 + sqrt(37) (where sqrt denotes the positive square root).

If you compute higher and higher powers of this number, they seem to get closer and closer to being integers
(i.e., the fractional part of the powers seems to be decreasing). Explain
why. Hint: this also happens for s = 6 + sqrt(35).