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

nicekikah 2021-03-09 Answered
Show that the congruence 8x2x+40(mod9) has no solution by reducing to the form y2a(mod9)
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Expert Answer

Yusuf Keller
Answered 2021-03-10 Author has 90 answers
Step 1
Given congruence is
8x2x+40(mod9)
We have to show this congruence has no solution by reducing it to the form y2=a(mod9)
Step 2
Consider the congruence
8x2x+40(mod9)
x2+8x+40(mod9)18(mod9)
x2+8x16+16+40(mod9) by adding and substracting 15
x2+8x16+200(mod9)
(x28x+15)+200(mod9)
(x4)2+200(mod9)
(x4)220(mod9)
(x4)220(mod9)gcd(1,9)=1, so cancellation law holds
(x4)22(mod9):02(mod9)
Take y = x-4, the we have the congruence
y22(mod9)
and the congruence y22(mod9) has no solution.
The given congruence has no solution.
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