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Fantezi Cebir => Fantezi Cebir => Konuyu başlatan: stuart clark - Ağustos 22, 2012, 09:47:35 ös

Başlık: Remainder
Gönderen: stuart clark - Ağustos 22, 2012, 09:47:35 ös

Find Remainder when 2008²⁰⁰⁷-2008 is divided by 2008²+2009

Başlık: Ynt: Remainder
Gönderen: proble_m - Ağustos 23, 2012, 11:02:45 ös

Since 2008²+2009 = 2008²+2008+1 and x³-1=(x-1)(x²+x+1) we can write
2008³=2007(2008²+2009)+1.
So,
2008²⁰⁰⁷-2008 =(2008³)⁶⁶⁹-2008 =  (2007(2008²+2009)+1)⁶⁶⁹-2008
now for mod(2008²+2009) this becomes
(2007(2008²+2009)+1)⁶⁶⁹-2008≡1-2008(mod(2008²+2009))
≡-2007(mod(2008²+2009))
≡-2007+2008²+2009(mod(2008²+2009))
≡2008²+2(mod(2008²+2009))

Başlık: Ynt: Remainder
Gönderen: stuart clark - Eylül 14, 2012, 12:33:48 ös

Thanks proble_m