Chứng tỏ rằng :A=1+5+52+53+....+597+598+599 chia hết cho 31
Giúp mình vs!!!
K
Khách
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Những câu hỏi liên quan
1 tháng 1 2017
A=1+5+52+533+.....+597+598+599
A=(1+5+52) +533×544×....×5599
A=31 +533×544×....×5599
A=31×533+544×...×5599
=> A ÷ 31
Theo mk nghi la vay . Hk chac nha
1 tháng 1 2017
đề bài khó quá, hay là bạn viết lộn đề rùi, dạng này mình chưa gặp bao giờ!
![YTP] Huấn Hoa Hồng Chế](data:image/jpeg;base64,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)
NM
4
BD
29 tháng 6 2023
0\(a.S=1-5+5^2-5^3+...+5^{98}-5^{99}\\ 5S=5-5^2+5^3-5^4+.....+5^{99}-5^{100}\\ 5S+S=\left(5-5^2+5^3-5^4+.....+5^{99}-5^{100}\right)+\left(1-5^{ }+5^2-5^3+.....+5^{98}-5^{99}\right)\\ 6S=1-5^{100}\\ S=\dfrac{1-5^{100}}{6}\\ \)
\(b,S6=1-5^{100}\\ 1-S6=5^{100}\)
=> 5100 chia 6 du 1
KL
2
OY
10 tháng 10 2021
Đặt \(A=1+5+5^2+5^3+...+5^{402}+5^{403}+5^{404}\)
\(\Rightarrow A=\left(1+5+5^2\right)+\left(5^3+5^4+5^5\right)+...+\left(5^{399}+5^{400}+5^{401}\right)+\left(5^{402}+5^{403}+5^{404}\right)\)
\(\Rightarrow A=31.1+31.5^3+...+31.5^{402}\)
\(\Rightarrow A=31\left(1+5^3+5^6+...+5^{402}\right)\)
\(\Rightarrow A⋮31\left(đpcm\right)\)
NP
5
DT
Dang Tung
CTVHS
23 tháng 9 2023
\(\left(1+5+5^2\right)+\left(5^3+5^4+5^5\right)+...+\left(5^{402}+5^{403}+5^{404}\right)\\ =31+5^3.\left(1+5+5^2\right)+...+5^{402}.\left(1+5+5^2\right)\\ =31+5^3.31+...+5^{402}.31\\ =31.\left(1+5^3+...+5^{402}\right)⋮31\left(DPCM\right)\)
GL
2
29 tháng 10 2023
\(B=5+5^2+5^3+...+5^{88}+5^{89}+5^{90}\)
\(=\left(5+5^2+5^3\right)+\left(5^4+5^5+5^6\right)+...+\left(5^{88}+5^{89}+5^{90}\right)\)
\(=5\left(1+5+5^2\right)+5^4\left(1+5+5^2\right)+...+5^{88}\left(1+5+5^2\right)\)
\(=31\left(5+5^4+...+5^{88}\right)⋮31\)
CM
21 tháng 12 2019
a, Ta có:
2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100
= 2 + 2 2 + 2 3 + 2 4 + 2 5 +...+ 2 96 + 2 97 + 2 98 + 2 99 + 2 100
= 2. 1 + 2 + 2 2 + 2 3 + 2 4 +...+ 2 96 1 + 2 + 2 2 + 2 3 + 2 4
= 2 . 31 + 2 6 . 31 + . . . + 2 96 . 31
= 2 + 2 6 + . . . + 2 96 . 31 chia hết cho 31
b, Ta có:
5 + 5 2 + 5 3 + 5 4 + 5 5 + 5 6 + . . . + 5 149 + 5 150
= 5 + 5 2 + 5 3 + 5 4 + 5 5 + 5 6 + . . . + 5 149 + 5 150
= 5 1 + 5 + 5 3 1 + 5 + 5 5 1 + 5 + . . . + 5 149 1 + 5
= 5 . 6 + 5 3 . 6 + 5 5 . 6 + . . . + 5 149 . 6
= ( 5 + 5 3 + 5 5 + . . . + 5 149 ) . 6 chia hết cho 6
Ta lại có:
5 + 5 2 + 5 3 + 5 4 + 5 5 + 5 6 + . . . + 5 149 + 5 150
= 5 + 5 2 + 5 3 + 5 4 + 5 5 + 5 6 +...+ 5 145 + 5 146 + 5 147 + 5 148 + 5 149 + 5 150 (có đúng 25 nhóm)
= [ ( 5 + 5 4 ) + ( 5 2 + 5 5 ) + ( 5 3 + 5 6 ) ] + ... + [ 5 145 + 5 148 ) + ( 5 146 + 5 149 ) + ( 5 147 + 5 150 ]
= [ 5 ( 1 + 5 3 ) + 5 2 ( 1 + 5 3 ) + 5 3 ( 1 + 5 3 ) ] + ... + [ 5 145 1 + 5 3 ) + 5 146 ( 1 + 5 3 ) + 5 147 ( 1 + 5 3 ]
= ( 5 . 126 + 5 2 . 126 + 5 3 . 126 ) + ... + ( 5 145 . 126 + 5 146 . 126 + 5 147 . 126 )
= ( 5 + 5 2 + 5 3 ) . 126 + ( 5 7 + 5 8 + 5 9 ) . 126 + ... + ( 5 145 + 5 146 + 5 147 ) . 126
= 126.[ ( 5 + 5 2 + 5 3 ) + ( 5 7 + 5 8 + 5 9 ) + ... + ( 5 145 + 5 146 + 5 147 ) ] chia hết cho 126.
Vậy 5 + 5 2 + 5 3 + 5 4 + 5 5 + 5 6 + . . . + 5 149 + 5 150 vừa chia hết cho 6, vừa chia hết cho 126
Ta thấy A gồm có 99 số hạng nên ta nhóm mỗi nhóm 3 số hạng.
Ta có: A = 1 + 5 + 52 + 53 + 54 + 55 +...+ 597 + 598 + 599
= (1 + 5 + 52 )+ (53 + 54 + 55 )+...+( 597 + 598 + 599 )
=(1 + 5 + 52 )+ 53(1 + 5 + 52 ) +...+ 597(1 + 5 + 52 )
= ( 1 + 5 + 52)(1 + 53+....+597)
= 31(1 + 53+....+597)
Vì có một thừa số là 31 nên A chia hết cho 31.
Cứ 3 số góp thành 1 nhóm: => A = (1+5+52) + (53+54+55) +...+(597+598+599) => A= 31 + 53(1+5+52) +...+ 597(1+5+52) => A= 31*(1+53+...+597) => A chia hết cho 31