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\(B=2+2^2+2^3+...+2^{60}\)
\(=2\left(1+2+2^2\right)+...+2^{58}\left(1+2+2^2\right)\)
\(=7\cdot\left(2+...+2^{58}\right)⋮7\)
a) 5+52+53+54+...+5100
= (5+52)+(53+54)+...+(599+5100)
= 30+52.(5+52)+...+598.(5+52)
= 30+52.30+...+598.30
= 30.(1+52+...+598)
Vì 30 chia hết cho 10
=> 30.(1+52+...+598) chia hết cho 10
=> 5+52+53+...+5100 chia hết cho 10
1)
a)\(B=3+3^3+3^5+3^7+.....+3^{1991}\)
\(\Leftrightarrow B=3\left(1+3^2+3^4+3^6+.....+3^{1990}\right)\)
Vì \(3\left(1+3^2+3^4+3^6+.....+3^{1990}\right)\)chia hết cho 3 nên \(B⋮3\)
\(B=3+3^3+3^5+3^7+.....+3^{1991}\)
\(\Leftrightarrow B=\left(3+3^3+3^5+3^7\right)+.....+\left(3^{1988}+3^{1989}+3^{1990}+3^{1991}\right)\)
\(\Leftrightarrow B=3\left(1+3^2+3^4+3^6\right)+.....+3^{1988}\left(1+3^2+3^4+3^6\right)\)
\(\Leftrightarrow B=3.820+.....+3^{1988}.820\)
\(\Leftrightarrow B=3.20.41+.....+3^{1988}.20.41\)
Vì \(3.20.41+.....+3^{1988}.20.41\) chia hết cho 41 nên \(B⋮41\)
2.
a,\(50-\left[\left(50-2^3.5\right):2+3\right]\)
\(=50-\left[\left(50-40\right):2+3\right]\)
\(=50-\left(10:2+3\right)\)
\(=50-8\)
\(=42\)
b,\(8697-\left[3^7:3^5+2\left(13-3\right)\right]\)
\(=8697-\left(3^2+2.10\right)\)
\(=8697-\left(9+20\right)\)
\(=8697-29\)
\(=8668\)
c,\(205-\left[1200-\left(4^2-2.3\right)^3\right]:40\)
\(=205-200:40\)
\(=200\)
2)
a) \(50-\left[\left(50-2^3.5\right):2+3\right]\)
\(=50-\left[\left(50-8.5\right):2+3\right]\)
\(=50-\left[\left(50-40\right):2+3\right]\)
\(=50-\left(10:2+3\right)\)
\(=50-\left(5+3\right)\)
\(=50-8\)
\(=42\)
b) \(8697-\left[3^7:3^5+2\left(13-3\right)\right]\)
\(=8697-\left(3^7:3^5+2.10\right)\)
\(=8697-\left(3^{7-5}+2.10\right)\)
\(=8697-\left(3^2+2.10\right)\)
\(=8697-\left(9+2.10\right)\)
\(=8697-\left(9+20\right)\)
\(=8697-29\)
\(=8668\)
c) \(205-\left[1200-\left(4^2-2.3\right)^3\right]:40\)
\(=205-\left[1200-\left(16-2.3\right)^3\right]:40\)
\(=205-\left[1200-\left(16-6\right)^3\right]:40\)
\(=205-\left(1200-10^3\right):40\)
\(=205-\left(1200-1000\right):40\)
\(=205-200:40\)
\(=205-5\)
\(=200\)