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# Serie arithmetique t

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### Serie arithmetique t

1. 1. ‫ﺍﻟﺠﺫﻉ ﺍﻟﻤﺸﺘﺭﻙ ﺍﻟﻌﻠﻤﻲ‬ ‫ﻭ ﻤﺒﺎﺩﺉ ﻓﻲ ﺍﻟﺤﺴﺎﺒﻴﺎﺕ‬ ‫ﺍﻟﻤﺠﻤﻭﻋﺔ‬ ‫ﺜﺎﻨﻭﻴﺔ ﺍﻟﻔﺘﺢ - ﺍﻟﺨﻤﻴﺴﺎﺕ‬ ‫‪Prof : A.BEN ELKHATIR‬‬ ‫ﺴﻠﺴﻠﺔ ﺭﻗﻡ 10‬ ‫8002/7002 ‪Année scolaire‬‬ ‫ﺗﻤﺮﻳﻦ40:‬ ‫ﺗﻤﺮﻳﻦ10:‬‫1(- ﻓﻜﻙ ﺇﻟﻰ ﺠﺩﺍﺀ ﻋﻭﺍﻤل ﺃﻭﻟﻴﺔ ﺍﻷﻋﺩﺍﺩ 704 ﻭ 385 ﻭ 1001 ، ﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺘﻔﻜﻴﻙ ﺍﻷﻋﺩﺍﺩ ﺍﻟﺘﺎﻟﻴﺔ :‬ ‫1(- ﺒﻴﻥ ﺃﻥ ﻤﺠﻤﻭﻉ ﺜﻼﺜﺔ ﺃﻋﺩﺍﺩ ﺼﺤﻴﺤﺔ ﻁﺒﻴﻌﻴﺔ ﻤﺘﺘﺎﺒﻌﺔ ﻴﻜﻭﻥ ﺩﺍﺌﻤﺎ ﻤﻀﺎﻋﻔﺎ ﻟﻠﻌﺩﺩ 3 .‬ ‫704704 = ‪ a‬ﻭ 385385 = ‪ b‬ﻭ ‪ a.b‬ﻭ ‪. a + b‬‬ ‫2(- ﺒﻴﻥ ﺃﻥ ﻤﺠﻤﻭﻉ ﺨﻤﺴﺔ ﺃﻋﺩﺍﺩ ﺼﺤﻴﺤﺔ ﻁﺒﻴﻌﻴﺔ ﻤﺘﺘﺎﺒﻌﺔ ﻴﻜﻭﻥ ﺩﺍﺌﻤﺎ ﻤﻀﺎﻋﻔﺎ ﻟﻠﻌﺩﺩ 5 .‬ ‫2(- ﻓﻜﻙ ﺇﻟﻰ ﺠﺩﺍﺀ ﻋﻭﺍﻤل ﺃﻭﻟﻴﺔ ﺍﻟﻌﺩﺩﻴﻥ 964 ﻭ 715 ، ﺜﻡ ﺃﻭﺠﺩ ﺠﻤﻴﻊ ﺃﺯﻭﺍﺝ ﺍﻷﻋﺩﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ‬ ‫3(- ﻫل ﻴﻤﻜﻥ ﻟﻤﺠﻤﻭﻉ ﺃﺭﺒﻌﺔ ﺃﻋﺩﺍﺩ ﺼﺤﻴﺤﺔ ﻁﺒﻴﻌﻴﺔ ﻤﺘﺘﺎﺒﻌﺔ ﺃﻥ ﻴﻜﻭﻥ ﻤﻀﺎﻋﻔﺎ ﻟﻠﻌﺩﺩ 4 ؟‬ ‫: ) 1‪( E‬‬ ‫ﺍﻟﻁﺒﻴﻌﻴﺔ ) ‪ ( x, y‬ﺤﻠﻭل ﻜل ﻤﻌﺎﺩﻟﺔ ﻤﻥ ﺍﻟﻤﻌﺎﺩﻟﺘﻴﻥ : 715 = ‪x − y‬‬ ‫2‬ ‫2‬ ‫4(- ﺒﻴﻥ ﺃﻥ ﺠﺩﺍﺀ ﺜﻼﺜﺔ ﺃﻋﺩﺍﺩ ﺼﺤﻴﺤﺔ ﻁﺒﻴﻌﻴﺔ ﻤﺘﺘﺎﺒﻌﺔ ﻴﻜﻭﻥ ﺩﺍﺌﻤﺎ ﻤﻀﺎﻋﻔﺎ ﻟﻠﻌﺩﺩ 6 ، ﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺃﻥ‬ ‫.‬ ‫: ) 2‪( E‬‬ ‫ﻭ 964 = ‪x − y‬‬ ‫3‬ ‫3‬ ‫ﺠﺩﺍﺀ ﺜﻼﺜﺔ ﺃﻋﺩﺍﺩ ﺼﺤﻴﺤﺔ ﻁﺒﻴﻌﻴﺔ ﺯﻭﺠﻴﺔ ﻭ ﻤﺘﺘﺎﺒﻌﺔ ﻴﻜﻭﻥ ﺩﺍﺌﻤﺎ ﻤﻀﺎﻋﻔﺎ ﻟﻠﻌﺩﺩ 84 .‬ ‫ﺗﻤﺮﻳﻦ50:‬ ‫ﺗﻤﺮﻳﻦ20:‬ ‫ﻨﻌﺘﺒﺭ ﺍﻟﻌﺩﺩ ﺍﻟﺼﺤﻴﺢ ﺍﻟﻁﺒﻴﻌﻲ ‪ a‬ﺒﺤﻴﺙ : 11 × 37 × 53 = ‪. a‬‬ ‫1(- ﺤﺩﺩ ﺠﻤﻴﻊ ﺃﺯﻭﺍﺝ ﺍﻷﻋﺩﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ ﺍﻟﻁﺒﻴﻌﻴﺔ ) ‪ ( x, y‬ﺤﻠﻭل ﻜل ﻤﻌﺎﺩﻟﺔ ﻤﻥ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ :‬‫1(- ﺃﻭﺠﺩ ﺃﺼﻐﺭ ﻋﺩﺩ ﺼﺤﻴﺢ ﻁﺒﻴﻌﻲ ﻏﻴﺭ ﻤﻨﻌﺩﻡ ‪ n‬ﻴﻜﻭﻥ ﻤﻥ ﺃﺠﻠﻪ ﺍﻟﻌﺩﺩ ‪ b = n.a‬ﻤﺭﺒﻌﺎ ﻜﺎﻤﻼ .‬ ‫: ) 3‪( E‬‬ ‫، 44 = 2 ‪9x 2 − 4 y‬‬ ‫: ) 2‪( E‬‬ ‫، 63 = 2 ‪x 2 − 4 y‬‬ ‫: ) 1‪( E‬‬ ‫61 = 2 ‪x 2 − y‬‬ ‫2(- ﺃﻭﺠﺩ ﺃﺼﻐﺭ ﻋﺩﺩ ﺼﺤﻴﺢ ﻁﺒﻴﻌﻲ ﻏﻴﺭ ﻤﻨﻌﺩﻡ ‪ m‬ﻴﻜﻭﻥ ﻤﻥ ﺃﺠﻠﻪ ﺍﻟﻌﺩﺩ ‪ c = m.a‬ﻤﻜﻌﺒﺎ ﻟﻌﺩﺩ‬ ‫ﺼﺤﻴﺢ ﻁﺒﻴﻌﻲ .‬ ‫.‬ ‫: ) 4‪( E‬‬ ‫ﻭ 2 ‪x 2 − 6 x + 54 = y‬‬ ‫ﺗﻤﺮﻳﻦ60:‬ ‫2(- ﺤﺩﺩ ﺠﻤﻴﻊ ﻗﻴﻡ ﺍﻟﻌﺩﺩ ﺍﻟﺼﺤﻴﺢ ﺍﻟﻁﺒﻴﻌﻲ ‪ n‬ﺍﻟﺘﻲ ﻴﻜﻭﻥ ﻤﻥ ﺃﺠﻠﻬﺎ ﻋﻠﻰ ﺍﻟﺘﻭﺍﻟﻲ :‬ ‫1(- ﺒﻴﻥ ﺃﻥ ﺍﻟﻌﺩﺩ 8888 + .... + 48 + 38 + 28 + 8 = ‪ A‬ﻴﻘﺒل ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ 37 .‬ ‫.‬ ‫: )3 (‬ ‫ﻭ 9 + 3‪n + 1| n‬‬ ‫: )2 (‬ ‫، 51 + ‪n + 7 | 2n‬‬ ‫: )1(‬ ‫5+‪n−2|n‬‬ ‫2(- ﻓﻜﻙ ﺇﻟﻰ ﺠﺩﺍﺀ ﻋﻭﺍﻤل ﺃﻭﻟﻴﺔ ﺍﻟﻌﺩﺩ 4371 ، ﺜﻡ ﺃﻭﺠﺩ ﺠﻤﻴﻊ ﺃﺯﻭﺍﺝ ﺍﻷﻋﺩﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ ﺍﻟﻁﺒﻴﻌﻴﺔ‬ ‫3(- ﺤﺩﺩ ﺠﻤﻴﻊ ﻗﻴﻡ ﺍﻟﻌﺩﺩ ﺍﻟﺼﺤﻴﺢ ﺍﻟﻁﺒﻴﻌﻲ ‪ n‬ﺍﻟﺘﻲ ﻴﻜﻭﻥ ﻤﻥ ﺃﺠﻠﻬﺎ ﻋﻠﻰ ﺍﻟﺘﻭﺍﻟﻲ :‬ ‫) ‪ ( x, y‬ﺍﻟﺘﻲ ﺘﺤﻘﻕ : 71 = ‪ x ∧ y‬ﻭ 4371 = ‪. xy‬‬ ‫3 + 2‪n‬‬ ‫52 + ‪3n‬‬ ‫71 + ‪n‬‬ ‫.‬ ‫: )3 (‬ ‫∈‬ ‫ﻭ‬ ‫: )2 (‬ ‫∈‬ ‫،‬ ‫: )1(‬ ‫∈‬ ‫.‬ ‫‪( E ) : ( x + 1)( y + 2 ) = 2 xy‬‬ ‫ﺒﺤﻴﺙ :‬ ‫2‬ ‫3(- ﺃﻭﺠﺩ ﺠﻤﻴﻊ ﺍﻷﺯﻭﺍﺝ ) ‪ ( x, y‬ﻤﻥ‬ ‫3−‪n‬‬ ‫3+‪n‬‬ ‫4−‪n‬‬ ‫ﺗﻤﺮﻳﻦ70:‬ ‫ﺗﻤﺮﻳﻦ30:‬ ‫ﻨﻌﺘﺒﺭ ﺍﻟﻌﺩﺩ ‪ N = mcdu‬ﺍﻟﻤﻜﻭﻥ ﻤﻥ ﺃﺭﺒﻌﺔ ﺃﺭﻗﺎﻡ .‬ ‫1(- ﺤﺩﺩ ﺠﻤﻴﻊ ﻗﻴﻡ ﺭﻗﻡ ﺍﻟﻭﺤﺩﺍﺕ ‪ x‬ﺍﻟﺘﻲ ﻴﻜﻭﻥ ﻤﻥ ﺃﺠﻠﻬﺎ ﺍﻟﻌﺩﺩ ‪ a = 54 x‬ﻤﻀﺎﻋﻔﺎ ل 6 .‬‫1(- ﺒﻴﻥ ﺃﻥ : ) ‪ ، N = ( m + c + d + u ) + 9 (100m + 10c + d‬ﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺃﻥ ‪ N‬ﻤﻀﺎﻋﻑ ل3‬ ‫2(- ﺤﺩﺩ ﺠﻤﻴﻊ ﻗﻴﻡ ﺭﻗﻡ ﺍﻟﻌﺸﺭﺍﺕ ‪ y‬ﺍﻟﺘﻲ ﻴﻜﻭﻥ ﻤﻥ ﺃﺠﻠﻬﺎ ﺍﻟﻌﺩﺩ 4 ‪ b = 53 y‬ﻤﻀﺎﻋﻔﺎ ل 3 ﻭ 4 .‬‫)ﻋﻠﻰ ﺍﻟﺘﻭﺍﻟﻲ ل9 ( ﺇﺫﺍ ﻭ ﻓﻘﻁ ﺇﺫﺍ ﻜﺎﻥ : ‪ m + c + d + u‬ﻤﻀﺎﻋﻑ ل3 )ﻋﻠﻰ ﺍﻟﺘﻭﺍﻟﻲ ل9 ( .‬ ‫3(- ﺒﻴﻥ ﺃﻨﻪ ﻤﻬﻤﺎ ﻴﻜﻥ ﺍﻟﺭﻗﻤﺎﻥ ‪ x‬ﻭ ‪ y‬ﺒﺤﻴﺙ ‪ x ≥ y‬ﻓﺈﻥ ﺍﻟﻌﺩﺩ ‪ b = xy − yx‬ﻤﻀﺎﻋﻑ ل 9 .‬‫2(- ﺒﻴﻥ ﺃﻥ : ‪ ، N = 100 (10m + c ) + du‬ﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺃﻥ ‪ N‬ﻤﻀﺎﻋﻑ ل4 )ﻋﻠﻰ ﺍﻟﺘﻭﺍﻟﻲ ل52 (‬ ‫4(- ﺒﻴﻥ ﺃﻨﻪ ﻤﻬﻤﺎ ﻴﻜﻥ ﺍﻟﺭﻗﻤﺎﻥ ‪ x‬ﻭ ‪ y‬ﻓﺈﻥ ﺍﻟﻌﺩﺩ ‪ c = xy + yx‬ﻤﻀﺎﻋﻑ ﻟﻌﺩﺩ ﺼﺤﻴﺢ ﻁﺒﻴﻌﻲ‬ ‫ﺇﺫﺍ ﻭ ﻓﻘﻁ ﺇﺫﺍ ﻜﺎﻥ ﺍﻟﻌﺩﺩ ‪ du‬ﻤﻀﺎﻋﻔﺎ ل 4 ) ﻋﻠﻰ ﺍﻟﺘﻭﺍﻟﻲ ل52 ( .‬ ‫ﻴﺨﺎﻟﻑ 1 ﻭ ﺤﺩﺩ ﻫﺫﺍ ﺍﻟﻌﺩﺩ .‬