x+{x+1}+{x+12}+{x+23}+{x+77}+{x+88}+[x+99}=370
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(12 + 77 + 34 + 23 + 88 + 45 + 55 + 66) x (45200 – 1808 x 25)
=(12 + 77 + 34 + 23 + 88 + 45 + 55 + 66) x (45200-45200)
=(12 + 77 + 34 + 23 + 88 + 45 + 55 + 66) x 0
=0
\(\dfrac{x-9}{55}+\dfrac{x-10}{66}=\dfrac{x-11}{77}+\dfrac{x-12}{88}\\ \Leftrightarrow\left(\dfrac{x-9}{55}+\dfrac{1}{11}\right)+\left(\dfrac{x-10}{66}+\dfrac{1}{11}\right)-\left(\dfrac{x-11}{77}+\dfrac{1}{11}\right)-\left(\dfrac{x-12}{88}+\dfrac{1}{11}\right)=0\)
\(\Leftrightarrow\left(\dfrac{x-9}{55}+\dfrac{5}{55}\right)+\left(\dfrac{x-10}{66}+\dfrac{6}{66}\right)-\left(\dfrac{x-11}{77}+\dfrac{7}{77}\right)-\left(\dfrac{x-12}{88}+\dfrac{8}{88}\right)=0\)
\(\Leftrightarrow\dfrac{x-4}{55}+\dfrac{x-4}{66}-\dfrac{x-4}{77}-\dfrac{x-4}{88}=0\)
\(\Leftrightarrow\left(x-4\right)\left(\dfrac{1}{55}+\dfrac{1}{66}-\dfrac{1}{77}-\dfrac{1}{88}\right)=0\\ \Leftrightarrow x=4\left(vì.\dfrac{1}{55}+\dfrac{1}{66}-\dfrac{1}{77}-\dfrac{1}{88}\ne0\right)\)
Ta có:
\(f\left(x\right)=x^{99}+x^{88}+x^{77}+...+x^{11}+1\)
\(f\left(x\right)=\left(x^{99}+x^{88}+x^{77}+...+x^{11}\right)+1\)
\(f\left(x\right)=\left[\left(x^9\right)^{11}+\left(x^8\right)^{11}+\left(x^7\right)^{11}+...+x^{11}\right]+1\)
Ta thấy:
\(\left(x^9\right)^{11}\) chia hết cho \(x^9\)
\(\left(x^8\right)^{11}\) chia hết cho \(x^8\)
\(..........\)
\(x^{11}\) chia hết cho \(x\)
\(1\) chia hết cho \(1\)
\(\Rightarrow f\left(x\right)\) chia hết cho \(g\left(x\right)\) ( Đpcm )
\(=\left[66+34+77+23+12+88+45+55\right]\times\left[1808\times\left(25-25\right)\right]\\ =0\)
\(x+\left(x+1\right)+\left(x+12\right)+\left(x+23\right)+\left(x+77\right)+\left(x+88\right)+\left(x+99\right)=370\)
\(\Leftrightarrow x+x+1+x+12+x+23+x+77+x+88+x+99=370\)
\(\Leftrightarrow7x+\left(1+12+23+77+88+99\right)=370\)
\(\Leftrightarrow7x+\left(1+99\right)+\left(12+88\right)+\left(23+77\right)=370\)
\(\Leftrightarrow7x+100+100+100=370\)
\(\Leftrightarrow7x+300=370\)
\(\Leftrightarrow7x=70\)
\(\Leftrightarrow x=10\)
Vậy \(x=10\)
\(x+x+x+x+x+x+x+1+99+12+88+23+77=370\)
\(7x+100+100+100=370\)
\(7x+300=370\)
\(7x=70\)
\(x=10\)