Tính bằng cách thuận tiện nhất ;
30/11 x 66/25 x75/210
(8/11+3/12)+3/11
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Ta có:
\(6:\frac{2}{3}=6:2:3=1\)
Vậy \(6:\frac{2}{3}=1\left(đpcm\right)\)
\(\frac{3}{x+5}=15\%\)\(\left(Đk:x\ne-5\right)\)
\(\Leftrightarrow\frac{3}{x+5}=\frac{3}{20}\)
\(\Leftrightarrow x+5=20\)
\(\Leftrightarrow x=20-5\)
\(\Leftrightarrow x=15\)\(\left(t/m\right)\)
Vậy\(x=15\)
Matcha
Xét m,n có 1 số chia hết cho 5 thì A \(⋮\)5
Xét m,n đều không chia hết cho 5
Ta có : với a \(⋮̸\)5 thì a có dạng : \(5k\pm1;5k\pm2\)
\(\Rightarrow a^4=\left(5k\pm1\right)^4=B\left(5\right)+1\)chia 5 dư 1
\(a^4=\left(5k\pm2\right)^4=B\left(5\right)+16=B\left(5\right)+1\)chia 5 dư 1
từ đó suy ra \(m^4\)chia 5 dư 1 ; \(n^4\)chia 5 dư 1
\(\Rightarrow m^4-n^4\)chia hết cho 5
\(\Rightarrow A⋮5\)
Vậy ....
Ta có: \(A=mn\left(m^4-n^4\right)=mn\left(m^4-1\right)-mn\left(n^4-1\right)\)
Xét \(a\left(a^4-1\right)=a\left(a^2-1\right)\left(a^2+1\right)=a\left(a^2-1\right)\left(a^2-4\right)+5a\left(a^2-1\right)\)
\(=a\left(a-1\right)\left(a+1\right)\left(a-2\right)\left(a+2\right)+5a\left(a^2-1\right)⋮5\)với mọi a nguyên bất kì
=> \(nm\left(m^4-1\right)=n\left[m\left(m^4-1\right)\right]⋮5\)với m nguyên
\(nm\left(m^4-1\right)=m\left[n\left(n^4-1\right)\right]⋮5\)với n nguyên
=> \(A=mn\left(m^4-n^4\right)=mn\left(m^4-1\right)-mn\left(n^4-1\right)\) chia hết cho 5.
Ta có:4=1+1+1+1=\(\frac{2009}{2009}+\frac{2010}{2010}+\frac{2011}{2011}+\frac{2008}{2008}\)
\(\frac{2008}{2009}+\frac{1}{2009}+\frac{2009}{2010}+\frac{1}{2010}+\frac{2010}{2011}+\frac{1}{2011}+\frac{2008}{2008}\)
Xét \(A=\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2008}\)
\(=\frac{2009}{2009}+\frac{2010}{2010}+\frac{2011}{2011}+\frac{2008}{2008}+\frac{1}{2008}+\frac{1}{2008}+\frac{1}{2008}\)
xét \(\frac{1}{2009}< \frac{1}{2008};\frac{1}{2010}< \frac{1}{2008};\frac{1}{2011}< \frac{1}{2008}\)
\(\Rightarrow4< A\)
\(\left(2+4+6+8+...+100\right)\times\left(15\times3-15\times2-15\right)\)
\(=\left[102\times50\div2\right]\times\left[15.\left(3-2-1\right)\right]\)
\(=2550\times0\)
\(=0\)
\(\frac{30}{11}\times\frac{66}{25}\times\frac{75}{210}\)
\(=\frac{30\times6\times11\times3\times25}{11\times25\times30\times7}\)
\(=\frac{18}{7}\)
\(\left(\frac{8}{11}+\frac{3}{12}\right)+\frac{3}{11}\)
\(=\frac{8}{11}+\frac{3}{12}+\frac{3}{11}\)
\(=\left(\frac{8}{11}+\frac{3}{11}\right)+\frac{3}{12}\)
\(=1+\frac{3}{12}\)
\(=\frac{5}{4}\)
Matcha
\(\frac{30}{11}\times\frac{66}{25}\times\frac{75}{210}\)
\(=\frac{30\times66\times75}{11\times25\times210}\)
\(=\frac{1\times6\times3}{1\times1\times7}\)
\(=\frac{18}{7}\)
\(\left(\frac{8}{11}+\frac{3}{12}\right)+\frac{3}{11}\)
\(=\frac{8}{11}+\frac{3}{12}+\frac{3}{11}\)
\(=1+\frac{3}{12}\)
\(=1+\frac{1}{4}\)
\(=\frac{5}{4}\)