Competition problem — Number theory

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Number theory Difficulty 5.4 AIME, harder Find the answer

For example, if $f(x)=x^{4}-1$ , then the congruence equation
$f(x) \equiv 0(\bmod 16)$

has eight solutions $x \equiv \pm 1, \pm 3, \pm 5, \pm 7(\bmod 16)$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

None

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.

Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". Here is the formatted output as requested:

None

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.