Để phân số \(\dfrac{m+7}{m-4}\) có gái trị là một số nguyên thì:

\(m+7⋮m-4\)

\(\Rightarrow\left(m-4\right)+11⋮m-4\)

Mà \(m-4⋮m-4\)

\(\Rightarrow11⋮m-4\Leftrightarrow m-4\inƯ\left(11\right)=\left\{\pm1;\pm11\right\}\)

Ta có bảng sau:

Vậy \(m\in\left\{-7;3;5;15\right\}\)

\(\left(1+\dfrac{7}{9}\right).\left(1+\dfrac{7}{20}\right).\left(1+\dfrac{7}{33}.\right)\left(1+\dfrac{7}{48}\right)...\left(1+\dfrac{7}{180}\right)\)

\(=\dfrac{16}{9}.\dfrac{27}{20}.\dfrac{40}{33}.\dfrac{55}{48}...\dfrac{7}{180}\)

\(=\dfrac{2.8}{1.9}.\dfrac{3.9}{2.10}.\dfrac{4.10}{3.11}.\dfrac{5.11}{4.12}...\dfrac{11.17}{10.18}\)

\(=\dfrac{\left(2.3.4.5...11\right).\left(8.9.10.11...17\right)}{\left(1.2.3.4...10\right).\left(9.10.11.12...18\right)}\)

\(=\dfrac{11.8}{1.18}=\dfrac{88}{18}=\dfrac{44}{9}\)

Gọi ƯCLN (2n+1,3n+2)=d

\(\Rightarrow2n+1⋮d\)

\(3n+2⋮d\)

\(\Rightarrow3n+2-2n+1⋮d\)

\(2\left(3n+2\right)-3\left(2n+1\right)⋮d\)

\(6n+4-6n+3⋮d\)

\(\Rightarrow1⋮d\Leftrightarrow d=1\)

Vậy ƯCLN \(\left(2n+1,3n+2\right)=1\Leftrightarrow\dfrac{2n+1}{3n+2}\) là p/s tối giản \(\left(dpcm\right)\)

Gọi ƯCLN (12n+1,30n+2) là d

\(\Rightarrow\left(12n+1\right)⋮d\)

\(\left(30n+2\right)⋮d\)

\(\Rightarrow5\left(12n+1\right)-2\left(30n+2\right)⋮d\)

\(\Rightarrow60n+5-60n-4⋮d\)

\(\Rightarrow1⋮d\Leftrightarrow d=1\)

Vậy ƯCLN \(\left(12n+1,30n+2\right)=1\Leftrightarrow\dfrac{12n+1}{30n+2}\) là p/s tối giản \(\left(dpcm\right)\)

câu hỏi của Nguyễn Thị Như Quỳnh

\(a,\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99.101}\)

\(=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...\dfrac{1}{99}-\dfrac{1}{101}\)

\(=1-\dfrac{1}{101}\)

\(=\dfrac{100}{101}\)

\(A=1-5-9+13+17-21-25+29+...+2001-2005-2009+2013+2017\)

\(=\left(1-5-9+13\right)+\left(17-21-25+29\right)+...+\left(2001-2005-2009+2013\right)+2017\)

\(=0+0+..+0+2017\)

\(=2017\)

\(What\) \(color\) \(..are..\) \(your\) \(eyes\)?

\(a,15x=-75\)

\(x=-75:15\)

\(x=-5\)

\(\Rightarrow x=-5\)

\(b,3\left|x\right|=18\)

\(\left|x\right|=18:3\)

\(\left|x\right|=6\)

\(\Rightarrow x=6\) hoặc \(x=-6\)