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Hint to puzzle 58: Fifth power plus five

Let d, a, and b be integers.  Use the result that if d | a (d divides a) and d | b, then d | ax + by, where x and y are arbitrary integers.

It may help to begin with a simpler example.
Show that the greatest common divisor of n2 + 4 and (n + 1)2 + 4 is equal to 17, if n congruent to 8 (mod 17), and equal to 1 otherwise.