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Question Number 110358 by bobhans last updated on 28/Aug/20

if positive integer x satisfies x^2 −4x+56 ≡14 (mod 17) , what is the minimum value of x.

$${if}\:{positive}\:{integer}\:{x}\:{satisfies}\:{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{56}\:\equiv\mathrm{14}\:\left({mod}\:\mathrm{17}\right)\: \\ $$$$,\:{what}\:{is}\:{the}\:{minimum}\:{value}\:{of}\:{x}. \\ $$

Answered by john santu last updated on 28/Aug/20

⇔x^2 −4x+4 + 52 = 14 (mod 17) ⇔ (x−2)^2 = −38 (mod 17 ) ⇔ (x−2)^2 = 13 (mod 17) now we need to see whether 13 is square modulo 17. ⇒ 13 = 13+ 3.17 = 64 (mod 17) then ⇔ (x−2)^2 =64 (mod 17) ⇔ (x−2)^2 = 8^2 (mod 17) { ((x−2=8 (mod 17))),((x−2=−8 (mod 17))) :} { ((x=10 (mod 17))),((x=11 (mod 17))) :} minimum value of x is 10

$$\Leftrightarrow{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{4}\:+\:\mathrm{52}\:=\:\mathrm{14}\:\left({mod}\:\mathrm{17}\right) \\ $$$$\Leftrightarrow\:\left({x}−\mathrm{2}\right)^{\mathrm{2}} \:=\:−\mathrm{38}\:\left({mod}\:\mathrm{17}\:\right) \\ $$$$\Leftrightarrow\:\left({x}−\mathrm{2}\right)^{\mathrm{2}} \:=\:\mathrm{13}\:\left({mod}\:\mathrm{17}\right) \\ $$$${now}\:{we}\:{need}\:{to}\:{see}\:{whether}\:\mathrm{13}\:{is}\: \\ $$$${square}\:{modulo}\:\mathrm{17}.\: \\ $$$$\Rightarrow\:\mathrm{13}\:=\:\mathrm{13}+\:\mathrm{3}.\mathrm{17}\:=\:\mathrm{64}\:\left({mod}\:\mathrm{17}\right) \\ $$$${then}\:\Leftrightarrow\:\left({x}−\mathrm{2}\right)^{\mathrm{2}} =\mathrm{64}\:\left({mod}\:\mathrm{17}\right) \\ $$$$\Leftrightarrow\:\left({x}−\mathrm{2}\right)^{\mathrm{2}} \:=\:\mathrm{8}^{\mathrm{2}} \:\left({mod}\:\mathrm{17}\right)\: \\ $$$$\:\begin{cases}{{x}−\mathrm{2}=\mathrm{8}\:\left({mod}\:\mathrm{17}\right)}\\{{x}−\mathrm{2}=−\mathrm{8}\:\left({mod}\:\mathrm{17}\right)}\end{cases} \\ $$$$\begin{cases}{{x}=\mathrm{10}\:\left({mod}\:\mathrm{17}\right)}\\{{x}=\mathrm{11}\:\left({mod}\:\mathrm{17}\right)}\end{cases} \\ $$$${minimum}\:{value}\:{of}\:{x}\:{is}\:\mathrm{10} \\ $$

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