exercices_arithm_trc by ayoubayoub

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									   ‫ز‬    ‫ا‬               ‫را‬      ‫ا‬                           ‫ا‬     ‫ذ:‬      ‫زا‬    ‫إ‬                     ‫ت‬           ‫:ا‬   ‫ر‬
                                                                                                  ‫م‬    ‫ك‬          ‫ع‬    ‫ى‬
                                                 ‫‪Chorfi_mouhsine@yahoo.fr‬‬
                                                                                                                ‫ر 1:‬
                                                                         ‫2و3و4و5و9.‬                    ‫أن ا د 02862‬     ‫1(‬
                                              ‫9≤‪. 0≤n‬‬             ‫اد 2 و 4 و 3 و 9‬         ‫ن ا د ‪n15n‬‬          ‫‪n‬‬      ‫2( د‬
                                                                                   ‫د9.‬         ‫أن ا د 72 + 7 × 5 × 63‬   ‫3(‬
                                                                                       ‫أن ا د 3 + 7 × 9 × 2 د دي .‬      ‫4(‬

                                                                                                                               ‫ا :‬
                                                  ‫2و5.‬                          ‫1( ** ر و ات ا د 02862 ه 0 إذن ا د 02862‬
       ‫3و9.‬                  ‫ت ا د 3 و 9 إذن ا د 02862‬                    ‫ع أر م ا د 02862 ه 81 = 0 + 2 + 8 + 6 + 2‬            ‫**‬
                  ‫4.‬                 ‫ت 4 إذن ا د 02862‬                   ‫ن ا د 02‬        ‫ات د 02862‬            ‫ات و ا‬    ‫ا‬   ‫** ر‬
                                                                                ‫2و3و4و5و9.‬                            ‫و ا د 02862‬
‫‪ n‬ه 0 أو 2 أو 4 أو 6 أو 8‬            ‫0و9‬        ‫ر‬          ‫أن ن ‪ n‬د زو‬              ‫تا د2‬                  ‫ن ا د ‪n15n‬‬          ‫2( **‬
        ‫رة‬        ‫تا د4ا‬            ‫ت 4 و أن‬                 ‫أن ن ا د ‪5n‬‬            ‫تا د4‬                  ‫ن ا د ‪n15n‬‬          ‫**‬
                                                                                 ‫‪ n‬ه 2 أو 6 .‬         ‫05 و 95 ه 25 و 65 ن‬
                                                                                                              ‫إذن 2 = ‪ n‬أو 6 = ‪. n‬‬
              ‫تا د9 .‬                ‫أن ن ا د 6 + ‪n + 1 + 5 + n = 2n‬‬             ‫تا د3و9‬                    ‫ن ا د ‪n15n‬‬          ‫**‬
                                                ‫ت9.‬                    ‫إذا آ ن 2 = ‪ n‬ن 01 = 6 + 2 × 2 = 6 + ‪n + 1 + 5 + n = 2n‬‬
                                        ‫ت 3 و9 .‬             ‫إذا آ ن 6 = ‪ n‬ن 81 = 6 + 21 = 6 + 6 × 2 = 6 + ‪n + 1 + 5 + n = 2n‬‬
                                                                                                              ‫ا د ‪ n‬ه 6.‬           ‫و‬
                                                   ‫‪ ) n = 9k‬آ (‬            ‫‪k‬‬         ‫د‬         ‫د 9 إذا آ ن‬             ‫3( ا د ‪ n‬ن‬
                                                      ‫3 × 9 + 4 × 3 × 7 × 5 × 41 × 3 = 72 + 21 × 7 × 5 × 24‬                     ‫:‬
                                                                            ‫3 × 9 + 4 × 7 × 5 × 41 × 3 × 3 =‬
                                                                            ‫3 × 9 + 4 × 7 × 5 × 41 × 9 =‬
                                                                            ‫)3 + 4 × 7 × 5 × 41( × 9 =‬
                                                 ‫‪Chorfi_mouhsine@yahoo.fr‬‬
                                                    ‫د9.‬              ‫‪n‬‬     ‫)3 + 4 × 7 × 5 × 41( = ‪ k‬و ‪ n = 9k‬و‬             ‫‪k‬‬       ‫و‬
                                                  ‫1 + ‪ ). n = 2k‬آ (‬              ‫‪k‬‬          ‫د‬          ‫أن‬       ‫د‬      ‫نا د ‪n‬‬       ‫4(‬
                                              ‫1 + ‪2 × 9 × 7 + 3 = 2 × ( 9 × 7 ) + 2 × 1 + 1 = 2 ( 9 × 7 ) + 1‬‬
                                                                                            ‫‪‬‬             ‫‪‬‬                     ‫:‬
                                          ‫دي .‬    ‫د‬     ‫‪n‬‬       ‫‪ k = ( 9 × 7 ) + 1‬و 1 + ‪ n = 2k‬و‬
                                                                                   ‫‪‬‬             ‫‪‬‬            ‫‪k‬‬            ‫د‬        ‫و‬

                                                                                                                ‫ر 2:‬
                                                                          ‫6462 = ‪ a‬و 0012 = ‪. b‬‬         ‫ا‬       ‫ا د ا‬
                                                                                      ‫اء ا أو .‬         ‫ا د ‪a‬و ‪b‬إ‬       ‫1(‬
                                                                                                                   ‫‪a‬‬
                                                                                                                 ‫.‬      ‫2(‬
                                                                                                                   ‫‪b‬‬
                                                                                                          ‫‪ a‬و ‪. b‬‬       ‫3(‬
                                                                                      ‫اء ا أو .‬        ‫ا د ‪ c=a b‬إ‬
                                                                                                             ‫2 3‬
                                                                                                                        ‫4(‬
                                                                                                                      ‫ا :‬
                                                                                                ‫ا د 6462 = ‪. a‬‬       ‫1(***‬
                                                                                                            ‫2 6462‬
                                                                                                            ‫3 3231‬
                                                                                                              ‫3 144‬
                                                                                                              ‫3 741‬
                                                                                                               ‫7 94‬
                                                                                                                 ‫7 7‬
                                                                                                                 ‫1‬
                                                                                    ‫2 7 × 33 × 2 = 7 × 7 × 3 × 3 × 3 × 2 = 4662 .‬       ‫و‬

                                                      ‫‪Riyadiyate.site.voila.fr‬‬
                                                                               ‫ا د 0012 = ‪b‬‬        ‫***‬
                                                                                       ‫2 0012‬
                                                                                       ‫2 0501‬
                                                                                         ‫3 525‬
                                                                                         ‫5 571‬
                                                                                          ‫5 53‬
                                                                                           ‫7 7‬
                                                                                           ‫1‬
                                                           ‫و 7 × 3 × 2 = 7 × 5 × 5 × 3 × 2 × 2 = 0012‬
                                                                                           ‫2‬

                                                                                             ‫‪a‬‬
                                                                                           ‫.‬        ‫2(‬
                                                                                             ‫‪b‬‬
                                                     ‫36 7 × 3 × 3 7 × 7 × 3 × 3 × 3 × 2 4662 ‪a‬‬
                                                      ‫=‬     ‫=‬                      ‫=‬          ‫=‬
                                                     ‫05 5 × 5 × 2 7 × 5 × 5 × 3 × 2 × 2 0012 ‪b‬‬
                                                                                   ‫‪ a‬و ‪. b‬‬          ‫3(‬
                            ‫** 6 12 = 7 × 3 3 × 2 = 2 7 × 33 × 2 = 2 7 × 33 × 2 = 4662 = ‪a‬‬
                                ‫** 12 2 = 7 × 3 × 2 = 7 × 3 × 22 = 7 × 3 × 22 = 0012 = ‪b‬‬
                           ‫: 2 7 × 23 × 42 × 6 7 × 93 × 32 = ) 7 × 3 × 22 ( × ) 2 7 × 33 × 2 ( = 2 ‪c = a 3b‬‬
                                                      ‫3‬                ‫2‬
                                                                                                              ‫4(‬
                                        ‫7× 3× 2 = 7× 7× 3× 3× 2 × 2 =‬
                                            ‫3‬    ‫4‬    ‫9‬     ‫2‬    ‫2‬    ‫6‬     ‫7‬    ‫11‬    ‫8‬

                                                                                               ‫ر 3:‬
                                                 ‫0041 = ‪ a‬و 0451 = ‪. b‬‬             ‫ا‬           ‫ا د ا‬
                                                             ‫اء ا أو .‬             ‫ا د ‪a‬و ‪b‬إ‬              ‫1(‬
                                                            ‫د ‪a‬و ‪.b‬‬             ‫كا آ‬         ‫ا‬       ‫2( أو ا‬
                                                        ‫د ‪a‬و ‪.b‬‬                ‫كا‬       ‫ا‬            ‫3( أو ا‬
                        ‫‪Chorfi_mouhsine@yahoo.fr‬‬
                                                                                                       ‫ا :‬
                                              ‫اء ا أو .‬           ‫ا د 0041 = ‪ a‬و 0451 = ‪ b‬إ‬               ‫1(‬
                      ‫2 0451‬                                                                ‫2 0041‬
                       ‫2 077‬                                                                 ‫2 007‬
                       ‫5 583‬                                                                 ‫2 053‬
                        ‫7 77‬                                                                 ‫5 571‬
                        ‫11 11‬                                                                  ‫5 53‬
                          ‫1‬                                                                      ‫7 7‬
                                                                                                 ‫1‬
                                                    ‫و : 7 × 5 × 2 = 7 × 5 × 5 × 2 × 2 × 2 = 0041 = ‪a‬‬
                                                                                          ‫3‬     ‫2‬


                                                    ‫11 × 7 × 5 × 2 2 = 11 × 7 × 5 × 2 × 2 = 0451 = ‪b‬‬       ‫و‬
                ‫أس .‬     ‫إ أ‬           ‫آ‬     ‫اء ا ا ا و ا‬        ‫د ‪a‬و ‪b‬ه‬              ‫كا آ‬         ‫ا‬    ‫2( ا‬
                  ‫أن : 7 × 5 × 2 = ‪ a‬و 11 × 7 × 5 × 2 = ‪ b‬ن 041 = 7 × 5 × 2 = ) ‪. PGCD ( a, b‬‬
                                     ‫2‬   ‫1‬                      ‫2‬     ‫1‬                   ‫3‬     ‫2‬


‫أآ أس .‬   ‫إ‬        ‫آ‬    ‫آ وا‬           ‫اء ا ا ا و ا‬      ‫د ‪a‬و ‪b‬ه‬              ‫كا‬     ‫ا‬       ‫3( ا‬
              ‫ان 7 × 5 × 2 = ‪ a‬و 11 × 7 × 5 × 2 = ‪ b‬ن 00451 = 11 × 7 × 5 × 2 = ) ‪. PPCM ( a, b‬‬
                                 ‫3‬   ‫2‬                        ‫2‬   ‫1‬                ‫3‬   ‫2‬




                                                                                        ‫ر 4:‬
                                                                                           ‫2+‪n‬‬
                           ‫.‬              ‫‪ n‬د‬        ‫تا د3‬              ‫أن ا د 5 − 5 = ‪A‬‬         ‫‪n‬‬
                                                                                                              ‫1(‬
                                                      ‫اء ا أو .‬           ‫ا د 53 × 301 = ‪ B‬إ‬                  ‫2(‬
                               ‫د 71 + ‪. n‬‬         ‫ن 4+‪n‬‬           ‫‪n‬‬          ‫ا‬      ‫ا دا‬     ‫د‬                ‫3(‬
                                                                                              ‫:‬                 ‫ا‬
                                                                        ‫.‬              ‫‪ n‬د‬                    ‫1(‬
                ‫) 5 × 8 ( × 3 = 5 × 42 = 42 × 5 = )1 − 52 ( 5 = )1 − 5 ( 5 = 5 − 5 = ‪A‬‬
                     ‫2+‪n‬‬ ‫‪n‬‬   ‫‪n‬‬    ‫2‬        ‫‪n‬‬            ‫‪n‬‬             ‫‪n‬‬             ‫‪n‬‬


                                                                 ‫‪A = 3k‬‬    ‫‪k‬‬                       ‫د‬          ‫و‬
                                                                      ‫تا د3.‬                     ‫:ا د ‪A‬‬        ‫و‬
                      ‫7 × 45 × 32 = 7 × 5 × 35 × 32 = 7 × 5 × ) 5 × 2 ( = 53 × 301 = ‪. B‬‬
                                                 ‫3‬
                                                                                                          ‫:‬   ‫2(‬
                            ‫‪Riyadiyate.site.voila.fr‬‬
                                   ‫د 31 .‬                  ‫4+‪n‬‬           ‫د 71 + ‪n‬‬                        ‫ن 4+‪n‬‬               ‫: 31 + ) 4 + ‪ n + 17 = ( n‬و‬                      ‫3(‬
                                                         ‫ن 1 = 4 + ‪ n‬أو 31 = 4 + ‪. n‬‬        ‫و أن ا ا د 31 ه : 1 و 31 و‬
                                                                        ‫.‬                ‫ن ‪ n‬د‬                 ‫ا د 1= 4 + ‪n‬‬
                            ‫د 71 + ‪ n‬ه 9 = ‪. n‬‬            ‫ن 4+‪n‬‬           ‫ا د ‪n‬‬            ‫وا ه 9 و‬          ‫ا د 31 = 4 + ‪n‬‬
                                                                                                                   ‫ر 5:‬
                                                                            ‫‪. a > 2b‬‬                            ‫‪a‬و ‪ b‬د‬
                                                                          ‫ا و .‬               ‫أن ا د ‪ a − 2b‬و ‪a + 2b‬‬      ‫1(‬
                                                                                     ‫‪ ℕ × ℕ‬ا د 63 = ‪. a − 4b‬‬
                                                                                         ‫2‬      ‫2‬
                                                                                                                          ‫2(‬
                                                                                                                        ‫ا :‬
                      ‫.‬   ‫ع ) ‪ ( a + 2b ) + ( a − 2b‬د زو‬      ‫نا‬     ‫: ‪ ( a + 2b ) + ( a − 2b ) = a + 2b + a − 2b = 2a‬و‬   ‫1(‬
                      ‫.‬   ‫ا و‬            ‫‪ a − 2b‬و ‪a + 2b‬‬             ‫إذن ا د‬                ‫د‬       ‫أو‬     ‫) ‪ ( a + 2b‬و ) ‪ ( a − 2b‬زو‬                 ‫نا د‬                     ‫و‬
                                                                                            ‫63 = ) ‪. ( a + 2b )( a − 2b‬‬                     ‫: 63 = ‪a − 4b‬‬
                                                                                                                                             ‫2‬        ‫2‬
                                                                                                                                                                              ‫2(‬
                   ‫ا د 63 ه : 1 و 2 و 3 و 4 و 6 و 9 و 21 و 81 و 63 .‬                            ‫أن ا‬         ‫د 63 و‬          ‫ن‬         ‫) ‪ ( a + 2b‬و ) ‪( a − 2b‬‬                ‫إذن‬
                     ‫81 = ‪a + 2b = 6 a + 2b‬‬
 ‫‪( a + 2b ≥ a − 2b ) ‬‬           ‫‪ ‬أو‬          ‫ن‬                           ‫و‬        ‫ا و‬                    ‫‪ a − 2b‬و ‪a + 2b‬‬              ‫ا د‬           ‫ال 1‬            ‫ا‬        ‫و‬
                     ‫2 = ‪ a − 2b = 6  a − 2b‬‬
                                                                                                              ‫81 = ‪a + 2b‬‬
                                                                                                              ‫‪‬‬                                       ‫ا‬           ‫ا‬           ‫**‬
                                                                                                              ‫2 = ‪ a − 2b‬‬
                             ‫81 = ‪a + 2b‬‬                   ‫81 = ‪a + 2b‬‬                     ‫‪‬‬      ‫81 = ‪a + 2b‬‬
         ‫01 = ‪a‬‬             ‫‪‬و‬                             ‫‪‬و‬                               ‫‪‬و‬                         ‫:‬                                  ‫ا‬               ‫ا‬
                             ‫01 = ‪ a‬‬                       ‫02 = ‪ 2a‬‬                        ‫2 + 81 = ‪a + 2b + a − 2b‬‬
                                                                                                ‫) 4 ,01(‬     ‫ه ا وج‬              ‫ا‬      ‫4 = ‪ . b‬إذن‬               ‫8 = ‪ 2b‬و‬
                                                                                                                             ‫6 = ‪a + 2b‬‬
                                                                 ‫) 0 ,6 ( .‬         ‫ا‬                             ‫ا‬          ‫‪‬‬                  ‫ا‬   ‫ا‬   ‫**‬
                                                                                                                             ‫6 = ‪ a − 2b‬‬
                                                        ‫‪ ℕ × ℕ‬ه ) 4 ,01( و ) 0 ,6 ( .‬                                              ‫أ ا ا د 63 = 2 ‪a 2 − 4b‬‬
                                                    ‫‪Chorfi_mouhsine@yahoo.fr‬‬
                                                                                                                                         ‫ر 6:‬
                                                                                                                  ‫م .‬                  ‫‪ n‬دا‬                                   ‫1(‬
                                                                                                                      ‫أن ‪ n 2 + n‬د زو .‬
                                                                                                                    ‫أن ا د 3 + ‪ n 2 + 5n‬د دي‬      ‫2(‬
                                                                                                                  ‫د4.‬        ‫أن ا د 2 ‪n 4 − n‬‬     ‫3(‬
                                                                                                                                                ‫ا :‬
                                                                                                                  ‫م .‬                 ‫‪ n‬دا‬        ‫1(‬
                                                                                                                             ‫: )1 + ‪n + n = n ( n‬‬
                                                                                                                              ‫2‬


                                                                                                                                             ‫1:‬     ‫ا‬
                                                             ‫.‬      ‫د زو‬           ‫اء )1 + ‪n ( n‬‬         ‫ا‬       ‫ن 1 + ‪ n‬د دي و‬    ‫إذا آ ن ‪ n‬دا زو‬
                                                                                                                                             ‫2:‬     ‫ا‬
                                                             ‫.‬       ‫د زو‬          ‫اء )1 + ‪n ( n‬‬         ‫ا‬       ‫إذا آ ن ‪ n‬دا د ن 1 + ‪ n‬د زو و‬
                                                                                        ‫.‬                    ‫د‬        ‫‪n‬‬              ‫د زو‬        ‫ن )1 + ‪n ( n‬‬                  ‫و‬
                                                                                                                 ‫: 3 + ‪n 2 + 5n + 3 = ( n 2 + n ) + 4 n‬‬                       ‫2(‬
                                                                                                                                                            ‫1:‬                  ‫ا‬
‫دي‬   ‫د‬    ‫‪(n‬‬   ‫2‬
                   ‫دي ن 3 + ‪+ n ) + 4n‬‬      ‫د‬       ‫أن 3‬    ‫و‬       ‫د زو‬           ‫‪(n‬‬   ‫2‬
                                                                                            ‫إذن ‪+ n ) + 4n‬‬            ‫د زو‬       ‫و ‪4n‬‬         ‫د زو‬        ‫أن ‪n + n‬‬
                                                                                                                                                              ‫2‬


                                                                                                                          ‫دي .‬   ‫د‬‫ن ا د 3 + ‪n 2 + 5n‬‬  ‫و‬
                                                                                                                                                  ‫2:‬   ‫ا‬
                                                ‫‪. n 2 + n = 2k‬‬                 ‫‪k‬‬                ‫د‬                 ‫ا ال 1(‬        ‫أن ‪ n + n‬د زو )‬‫2‬


                                                                                                                  ‫ن ) ‪. 4n = 2 ( 2n‬‬       ‫و ا د ‪ 4n‬د زو‬
                                                                                                                                            ‫1+ 2 = 3 .‬                ‫وا د3‬
                          ‫1 + )1 + ‪n + 5n + 3 = ( n + n ) + 4n + 3 = 2k + 2 ( 2n ) + 2 + 1 = 2 ( k + 2n‬‬
                            ‫2‬                   ‫2‬
                                                                                                                                                                        ‫إذن :‬
                                                       ‫‪Riyadiyate.site.voila.fr‬‬
                   . n 2 + 5n + 3 = 2k '+ 1    k' ‫د‬            ‫ د دي‬n 2 + 5n + 3 ‫و ا د‬
                        .             ‫ د‬k        a = 4k ‫د 4 إذا آ ن‬        a ‫3( ن ا د‬
                                               n − n = ( n − n )( n + n )
                                                 4    2    2        2
                                                                                   :
. n 2 + n = 2k       k            ‫د‬        ‫ د زو إذن‬n 2 + n ‫ا ال ا ول أن ا د‬                ‫و‬
           . n − n = 2k '
              2
                              k'             ‫د‬      ‫ د زو إذن‬n − n ‫ا د‬
                                                                    2
                                                                                     ‫ا‬      ‫و‬
      .4‫تا د‬              n − n2 ‫ا د‬
                           4
                                        ‫ و‬n − n = ( n − n )( n + n ) = 2k × 2k ' = 4kk ' : ‫إذن‬
                                           4   2     2        2




                   Riyadiyate.site.voila.fr

								
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