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if-positive-integer-x-satisfies-x-2-4x-56-14-mod-17-what-is-the-minimum-value-of-x-

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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.
ifpositiveintegerxsatisfiesx2−4x+56≡14(mod17),whatistheminimumvalueofx.
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
⇔x2−4x+4+52=14(mod17)⇔(x−2)2=−38(mod17)⇔(x−2)2=13(mod17)nowweneedtoseewhether13issquaremodulo17.⇒13=13+3.17=64(mod17)then⇔(x−2)2=64(mod17)⇔(x−2)2=82(mod17){x−2=8(mod17)x−2=−8(mod17){x=10(mod17)x=11(mod17)minimumvalueofxis10

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