Modular Arithmetic with Algebra

Question

If positive integer $x$ satisfies $x^2 - 4x +56 \equiv 14\pmod{17}$, find the minimum value of $x$.

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I have a solution that uses the quadratic formula, but I'm looking for more elegant ways to arrive on the answer ($10$).

Answer 1

This may be more elegant than the quadratic formula, but still uses some simple plugging-and-chugging.

Rearranging gets $x^2-4x+42\equiv 0\pmod{17}$, and changing to vertex form gets $$(x-2)^2+38\equiv 0\pmod{17}.$$

Now, $$(x-2)^2\equiv -38 \pmod{17} \rightarrow (x-2)^2\equiv -4\pmod{17} \rightarrow (x-2)^2\equiv 13\pmod{17}.$$

From here, just plug-and-chug. Since the RHS must be a perfect square along with the LHS, I would just list the positive integers that are $13\pmod{17}$.

$$13, 30, 47, \boxed{64}$$

So $x = 10.$

Answer 2

$x^2-4x+42\equiv0\bmod17$

$x^2+13x+42\equiv0\bmod17$

$(x+6)(x+7)\equiv0\bmod17$

$x\equiv-6$ or $-7\bmod17$

$x\equiv11$ or $10\bmod17$