子洋 asked in 科學數學 · 8 years ago

餘數問題23^461 is divided by 43

想請問23^461 is divided by 43

這題的答案是多少~~??

2 Answers

Rating
  • Lv 5
    8 years ago
    Best Answer

    餘數問題23^461 is divided by 43想請問23^461 is divided by 43

    這題的答案是多少~~??

    以[N]表示: N 除以43的餘數

    [23^461]=[23*23^460]=[23]*[23^460]=]=[23]*[(23^2]=[23]*[(529)^230] = [23]*[(12*43+13)^230]= [23]*[13^230] = [23]*[(13^2)^115] = [23]*[(3*43+40)^115] = [23]*[40^115] = [23]*[(40*40^114] = [23]*[40]*[40^114] = [23*40]*[(40^2)^57]= [920]*[(37*43+9)^57]=[21*43+17]*[9^57]= [17]*[9^57] = [17]*[9^57]= [17]*[(9^3)^19]=[17]*[(729)^19]=[17]*[(16*43+41)^19]=[17]*[41^19]=[17]*[41*41^18]=[17]*[41]*[41^18]=[17*41]*[(41^2)^9]=[16*43+9]*[(39*43+4)^9]=[9]*[4^9]=[9]*[64^3]=[9]*[(43+21)^3]=[9]*[21^3]=[9]*[215*43+16]=[9]*[16]=[9*16]=[144]= 15 ANS

    這題的答案是 15

    2012-05-31 19:24:13 補充:

    上傳亂掉

    更正:

    [23^461]=[23*23^460]=[23]*[23^460]=[23]*[(23^2)^230]=[23]*[(529)^230] = [23]*[(12*43+13)^230]= [23]*[13^230] =

  • 8 years ago

    由費馬小定理,23^42 ≡ 1,即得 23^461 ≡ 23^41 (mod 43)。

    題目等於在找 23 (mod 43) 的逆元。

    因為 23(15) - (43)(8) = 1,故 (23)(15) ≡ 1 (mod 43)。

    結論:23^461 ≡ 15 (mod 43)。

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