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\(\dfrac{6^5.\left(-12\right)^6}{\left(-4\right)^9\left(-3\right)^{10}}\)

\(=\dfrac{6^5.12^6}{\left(-4\right)^9.3^{10}}\)

\(=\dfrac{2^5.3^5.2^{12}.3^6}{\left(-1\right).2^{18}.3^{10}}\)

\(=\dfrac{2^{17}.3^{11}}{\left(-1\right).2^{18}.3^{10}}\)

\(=\dfrac{3}{\left(-1\right).2}\)

\(=\dfrac{-3}{2}\)

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Gọi số đó là a. (\(100000\le a\le999999\))

Theo bài ra, ta có:

a chia 2009 dư 209, a nhỏ nhất, a có 6 chữ số

\(\Rightarrow a-209⋮2009\)

\(\Rightarrow a-209\in B\left(2009\right)\)

\(\Rightarrow a-209\in\left\{2009;4018;...;98441;100450;\right\}\)

\(\Rightarrow a\in\left\{2218;4227;...;98650;100659;\right\}\)

=> a=100659 là nhỏ nhất thỏa mãn \(100000\le a\le999999\)

Vậy số tự nhiên nhỏ nhất có 6 chữ số mà chia cho 2009 dư 209 là 100659.

a, Ta có:

\(50< 2^n< 100\)

\(\Rightarrow2^5+18< 2^n< 2^6+36\)

\(\Rightarrow5< n\le6\) (Vì \(2^6+36>2^6\))

\(\Rightarrow n=6\)

Vậy \(n=6\).

b, Ta có:

\(50< 7^n< 2500\)

\(\Rightarrow7^2+1< 7^n< 7^4+99\)

\(\Rightarrow2< n\le4\) (Vì \(7^4+99>7^4\))

\(\Rightarrow n\in\left\{3;4\right\}\)

Vậy \(n\in\left\{3;4\right\}\)

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+, Ta có:

\(B=23!+19!-15!\)

\(B=\left(1\times2\times...\times11\times...\times23\right)+\left(1\times2\times...\times11\times...\times19\right)-\left(1\times2\times...\times11\times...\times15\right)\)

\(B=11\times\left[\left(1\times2\times...\times10\times12\times...\times23\right)+\left(1\times2\times...\times10\times12\times...\times19\right)-\left(1\times2\times...\times10\times12\times...\times15\right)\right]\)

\(\Rightarrow B⋮11\)

+, Ta có:

\(B=23!+19!-15!\)

\(B=\left(1\times2\times...\times10\times11\times...\times23\right)+\left(1\times2\times...\times10\times11\times...\times19\right)-\left(1\times2\times...\times10\times11\times...\times15\right)\)

\(B=11\times10\times\left[\left(1\times2\times...\times9\times12\times...\times23\right)+\left(1\times2\times...\times9\times12\times...\times19\right)-\left(1\times2\times...\times9\times12\times...\times15\right)\right]\)

\(B=110\times\left[\left(1\times2\times...\times9\times12\times...\times23\right)+\left(1\times2\times...\times9\times12\times...\times19\right)-\left(1\times2\times...\times9\times12\times...\times15\right)\right]\)

\(\Rightarrow B⋮110\)

+,Ta có:

\(B=23!+19!-15!\)

\(B=\left(1\times2\times...\times5\times...\times23\right)+\left(1\times2\times...\times5\times...\times19\right)-\left(1\times2\times...\times5\times...\times15\right)\)

\(B=5\times\left[\left(1\times2\times...\times4\times6\times...\times23\right)+\left(1\times2\times...\times4\times6\times...\times19\right)-\left(1\times2\times...\times4\times6\times...\times15\right)\right]\)

\(\Rightarrow B⋮5\)

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