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Serie arithmetique t
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 .
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