Show that the congruence 8x^2 -x + 4 -= 0(mod 9) has no solution by reducing to the form y^2 -= a(mod 9)

Question

Show that the congruence

8 x^{2} - x + 4 \equiv 0 (mod 9)

has no solution by reducing to the form

y^{2} \equiv a (mod 9)

Answer 1

Step 1
Given congruence is

8 x^{2} - x + 4 \equiv 0 (mod 9)

We have to show this congruence has no solution by reducing it to the form

y^{2} = a (mod 9)

Step 2
Consider the congruence

8 x^{2} - x + 4 \equiv 0 (mod 9)

\Rightarrow - x^{2} + 8 x + 4 \equiv 0 (mod 9) ∵ - 1 \equiv 8 (mod 9)

\Rightarrow - x^{2} + 8 x - 16 + 16 + 4 \equiv 0 (mod 9)

by adding and substracting 15

\Rightarrow - x^{2} + 8 x - 16 + 20 \equiv 0 (mod 9)

\Rightarrow - (x^{2} - 8 x + 15) + 20 \equiv 0 (mod 9)

\Rightarrow - {(x - 4)}^{2} + 20 \equiv 0 (mod 9)

\Rightarrow - {(x - 4)}^{2} \equiv 20 (mod 9)

\Rightarrow {(x - 4)}^{2} \equiv 20 (mod 9) ∵ gcd (- 1, 9) = 1

, so cancellation law holds

\Rightarrow {(x - 4)}^{2} \equiv 2 (mod 9) : 0 \equiv 2 (mod 9)

Take y = x-4, the we have the congruence

y^{2} \equiv 2 (mod 9)

and the congruence

y^{2} \equiv 2 (mod 9)

has no solution.

\Rightarrow

The given congruence has no solution.

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