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a
Ta có:\(2020\equiv1\left(mod3\right)\Rightarrow2020^{2019}\equiv1\left(mod3\right)\Rightarrow2020^{2019}-1\equiv0\left(mod3\right)\)
Khi đó:\(\left(2020^{2019}+1\right)\cdot\left(2020^{2019}-1\right)\equiv0\left(mod3\right)\)
suy ra đpcm
b
\(n^5+96n=n\left(n^4+96\right)\)
Để \(n^5+96n\) là số nguyên tố thì:\(n^4+96=1\left(h\right)n=1\)
Do \(n^4+96>1\Rightarrow n=1\)
Thay vào ta thấy thỏa mãn
Vậy n=1
a, =2020^4038 -1
Vì \(2020 \equiv 1 \pmod{3}\)
->\(2020^(4038) \equiv 1 \pmod{3}\)
->2020^4038 -1 chia hết cho 3 -> dpcm
a) \(2\left(\dfrac{2}{3.5}+\dfrac{4}{5.9}+...+\dfrac{16}{n\left(n+16\right)}\right)=\dfrac{16}{25}\)
\(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+...+\dfrac{1}{n}-\dfrac{1}{n+16}=\dfrac{8}{25}\)
\(\dfrac{1}{3}-\dfrac{1}{n+16}=\dfrac{8}{25}\)
\(\dfrac{n+13}{3\left(n+16\right)}=\dfrac{8}{25}\)
\(24n+384=25n+325\)
\(25n-24n=384-325\)
\(n=59\)
a)\(M=\frac{2019\times2020-2}{2018+2018\times2020}=\frac{2019\times2020-2}{2018+2018\times2020+2020-2020}=\frac{2019\times2020-2}{\left(2018+1\right)\times2020+2018-2020}=\frac{2019\times2020-2}{2019\times2020-2}=1\\ N=\frac{-2019\times20202020}{20192019\times2020}=\frac{-2019\times10001\times2020}{2019\times10001\times2020}=-1\)
b)\(5\left|x-1\right|=3M-2N=5\\ \left|x-1\right|=1\Rightarrow\hept{\begin{cases}x-1=1\Rightarrow x=2\\x-1=-1\Rightarrow x=0\end{cases}}\)
=>\(2\left(\dfrac{1}{2}+\dfrac{1}{6}+...+\dfrac{1}{n\left(n+1\right)}\right)=\dfrac{4039}{2020}\)
=>\(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n}-\dfrac{1}{n+1}=\dfrac{4039}{4040}\)
=>1-1/(n+1)=4039/4040
=>1/(n+1)=1/4040
=>n+1=4040
=>n=4039