Competition problem — Number theory

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Number theory Difficulty 5.4 AIME, harder Find the answer

Example 1 Find the solution to the congruence equation $4 x^{2}+27 x-12 \equiv 0(\bmod 15)$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Solve the absolute minimal complete residue system modulo 15: $-7,-6, \cdots,-1,0,1$ , $2, \cdots, 7$ . Direct calculation shows that $x=-6,3$ are solutions. Therefore, the solutions to this congruence equation are
$x \equiv-6,3(\bmod 15),$

The number of solutions is 2.

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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.