tìm dư 5^2014 + 47 chia cho24
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có: \(5^{2014}=\left(5^2\right)^{1007}=25^{1007}\)
Mà\(25\equiv1\left(mod24\right)\)
\(\Rightarrow25^{1007}\equiv1\left(mod24\right)\)
Lại có: \(47\equiv-1\left(mod24\right)\)
\(\Rightarrow25^{1007}+47\equiv0\left(mod24\right)\)
\(\Rightarrow5^{2015}+47\)chia cho 24 dư 0
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có:
\(2016\equiv0\left(mod24\right)\Rightarrow2016^{2017}\equiv0\left(mod24\right)\)
\(2017\equiv1\left(mod24\right)\Rightarrow2017^{2016}\equiv1^{2016}=1\left(mod24\right)\)
\(\Rightarrow2016^{2017}+2017^{2016}\equiv0+1=1\left(mod24\right)\)
Vậy số dư trong phép chia 20162017 + 20172016 cho 24 là 1
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Lời giải:
$A=5+5^2+5^3+(5^4+5^5+5^6+5^7)+(5^8+5^9+5^{10}+5^{11})+...+(5^{2012}+5^{2013}+5^{2014}+5^{2015})$
$=(1+5+5^2+5^3)+5^4(1+5+5^2+5^3)+5^8(1+5+5^2+5^3)+...+5^{2012}(1+5+5^2+5^3)-1$
$=(1+5+5^2+5^3)(1+5^4+5^8+...+5^{2012})-1$
$=156(1+5^4+...+5^{2012})-1$
$=13.12(1+5^4+...+5^{2012})-1$
$\Rightarrow A$ chia $13$ dư $-1$
Hay $A$ chia $13$ dư $12$
![](https://rs.olm.vn/images/avt/0.png?1311)
Áp dụng :An - Bn = (A-B)( An-1+An-2B +...+ ABn-2+Bn-1
=> 52014 +47 = (52014 - 1) +48
=(52 -1) (51006 +...+1) +48
= 24(51006 +...+1) +48
Vì 24 , 48 chia hết cho 12
=> 52014 + 47 chia hết cho 12