منتديات الرياضيات العربية - عرض مشاركة واحدة

22-03-2008, 02:14 AM

نجد بعض القيم لباقي القسمة على $100$ .

نرمز ل mod ب $\equiv$ و نضع $X= 2008$

لدينا ادن

$X\equiv 8 [100]\\X^2\equiv 64 [100]\\X^3\equiv 12 [100]\\X^4\equiv96 [100]\\X^5\equiv 68 [100]\\X^6\equiv 44 [100]\\X^7\equiv 52 [100]\\X^8\equiv 16 [100]\\X^9\equiv 28 [100]\\X^{10}\equiv 24 [100]\\X^{11}\equiv 92 [100]\\X^{12}\equiv 36 [100]\\X^{13}\equiv 88 [100]$

$\\X^{14}\equiv 4 [100]\\X^{15}\equiv 32 [100]\\X^{16}\equiv 56 [100]\\X^{17}\equiv 48[100]\\X^{18}\equiv 84 [100]\\X^{19}\equiv 72[100]\\X^{20}\equiv 76 [100]\\X^{21}\equiv 8 [100]\\X^{22}\equiv 64 [100]\\X^{23}\equiv 12 [100]\\X^{24}\equiv 96 [100]\\X^{25}\equiv 68 [100]\\X^{26}\equiv 44 [100]\\X^{27}\equiv 52 [100]\\X^{28}\equiv 16 [100]$

$\\X^{29}\equiv 28 [100]\\X^{30}\equiv 24 [100]\\X^{31}\equiv 92 [100]\\X^{32}\equiv 36[100]\\X^{33}\equiv 88 [100]\\X^{34}\equiv 4[100]\\X^{35}\equiv 32 [100]\\X^{36}\equiv 56 [100]\\X^{37}\equiv 48 [100]\\X^{38}\equiv 84 [100]\\X^{39}\equiv 72 [100]\\X^{40}\equiv 76 [100]\\X^{41}\equiv 8[100]\\X^{42}\equiv 64 [100]\\X^{43}\equiv 12[100]\\X^{44}\equiv 96[100]\\X^{45}\equiv 68[100]\\X^{46}\equiv 44 [100]\\X^{47}\equiv 52[100]\\X^{48}\equiv 16 [100]\\X^{49}\equiv 28[100]$

نلاحض انه ادا كان الاس اي P يكتب على الشكل $P=20K+8$ حيت K منZ فان الباقي هو $r=16$
و بماان $2008=20\times100+8$ فان باقي القسمة الاقليدية ل $2008^{2008}$ على $100$ هو $16$

اعتمدت على ان

$2008^8\equiv16[100]\\2008^{28}\equiv16[100]\\2008^{48}\equiv16[100]$

ارجو ان يكون جوابي صحيحا