a) \(A=2+2^2+2^3+2^4+.....+2^{100}\)
\(\Leftrightarrow A=\left(2+2^2\right)+\left(2^3+2^4\right)+.....+\left(2^{99}+2^{100}\right)\)
\(\Leftrightarrow A=2\left(1+2\right)+2^3\left(1+2\right)+.....+2^{99}\left(1+2\right)\)
\(\Leftrightarrow A=2.3+2^3.3+.....+2^{99}.3\)
\(\Leftrightarrow A=3\left(2+2^3+.....+2^{99}\right)⋮3\left(dpcm\right)\)
b) \(A=2+2^2+2^3+2^4+.....+2^{100}\)
\(\Leftrightarrow A=\left(2+2^2+2^3+2^4\right)+.....+\left(2^{97}+2^{98}+2^{99}+2^{100}\right)\)
\(\Leftrightarrow A=2\left(1+2+2^2+2^3\right)+.....+2^{97}\left(1+2+2^2+2^3\right)\)
\(\Leftrightarrow A=2.15+2^5.15+.....+2^{97}.15\)
\(\Leftrightarrow A=15\left(2+2^5+.....+2^{97}\right)\)
\(\Leftrightarrow A=3.5\left(2+2^5+.....+2^{97}\right)⋮5\left(dpcm\right)\)
2) \(A=2+2^2+2^3+2^4+....+2^{100}\)
\(\Leftrightarrow A=2\left(1+2+2^2+2^3+2^4+....+2^{99}\right)\)
\(\Leftrightarrow A=2\left[\left(1+2+2^2+2^3\right)+....+\left(2^{96}+2^{97}+2^{98}+2^{99}\right)\right]\)
\(\Leftrightarrow A=2\left[\left(1+2+2^2+2^3\right)+....+2^{96}\left(1+2+2^2+2^3\right)\right]\)
\(\Leftrightarrow A=2\left(15+....+2^{96}.15\right)\)
\(\Leftrightarrow A=2.15\left(1+2^4+2^8....+2^{92}+2^{96}\right)\)
\(\Leftrightarrow A=30\left(1+2^4+2^8....+2^{92}+2^{96}\right)\)
\(\Leftrightarrow A=....................0\)
Vậy chữ số tận cùng của A là 0
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