Toán Lớp 6: Cho A= 2+2^2+2^2+-+2^20 .Chứng tỏ rằng A chia hết cho 2,3,5

Question

Toán Lớp 6:  Cho A= 2+2^2+2^2+....+2^20 .Chứng tỏ rằng A chia hết cho 2,3,5

huongtructram · Answer

a)A = 2 + 2^2 + 2^3 +............+ 2^20  A = 2 + 2 . 2 + 2 . 2^2 +................+ 2 . 2^19   A = 2.( 1 + 2 + 2^2 +............+ 2^19 )  V&igrave; 2vdots2  => 2.( 1 + 2 + 2^2 +................+ 2^19 )vdots2  =>Avdots2  b)A = 2 + 2^2 + 2^3 + .. + 2^20  A = ( 2 + 2^2 ) + ( 2^3 + 2^4 ) +...............+ ( 2^19 + 2^20 )  A = 2.( 1 + 2 ) + 2^3.( 1 + 2 ) +............+ 2^19.( 1 + 2 )  A = 2 . 3 + 2^3 . 3 +............+ 2^19 . 3  A = 3.( 2 + 2^3 +...........+ 2^19 )  V&igrave; 3vdots3  =>3.( 2 + 2^3 +.........+ 2^19 )vdots3  =>Avdots3  c) A = 2 + 2^2 + 2^3 + .. + 2^20  A = ( 2 + 2^2 + 2^3 + 2^4 ) +............ + ( 2^17 + 2^18 + 2^19 + 2^20 )  A = 2.( 1 + 2 + 2^2 + 2^3 ) +...........+ 2^17.( 1 + 2 + 2^2 + 2^3 )  A = 2 . 15 +...........+ 2^17 . 15  A = 15.( 2 +..........+ 2^17 )  V&igrave; 15vdots15  =>15.( 2 +..........+ 2^17 )  =>Avdots15

tonhitruc · Answer

a , A = 2 + 2^2 + 2^3 + .. + 2^20       A = 2 + 2 . 2 + 2 . 2^2 + .. + 2 . 2^19        A = 2 ( 1 + 2 + 2^2 + .. + 2^19 )       &hArr; A chia hết cho 2       b , A = 2 + 2^2 + 2^3 + .. + 2^20       A = ( 2 + 2^2 ) + ( 2^3 + 2^4 ) + .. + ( 2^19 + 2^20 )       A = 2 ( 1 + 2 ) + 2^3 ( 1 + 2 ) + .. + 2^19 ( 1 + 2 )       A = 2 . 3 + 2^3 . 3 + .. + 2^19 . 3       A = ( 2 + 2^3 + .. + 2^19 ) . 3       &hArr; A chia hết cho 3       c , A = 2 + 2^2 + 2^3 + .. + 2^20       A = ( 2 + 2^2 + 2^3 + 2^4 ) + ( 2^5 + 2^6 + 2^7 + 2^8 ) + .. + ( 2^17 + 2^18 + 2^19 + 2^20 )       A = 2 ( 1 + 2 + 2^2 + 2^3 ) + 2^5 ( 1 + 2 + 2^2 + 2^3 ) + .. + 2^17 ( 1 + 2 + 2^2 + 2^3 )       A = 2 . 15 + 2^5 . 15 + .. + 2^17 . 15       A = ( 2 + 2^5 + .. + 2^17 ) . 15       &hArr; A chia hết cho 5