Bài 1: Tính
\(a)\dfrac{5}{6}+6\dfrac{5}{6}.\left(11\dfrac{94}{1591}-6\dfrac{38}{1517}\right):8\dfrac{11}{43}\)
\(b)1\dfrac{1}{6}+6\dfrac{5}{6}.\left(10\dfrac{133}{2173}-5\dfrac{23}{1643}\right):41\dfrac{12}{31}\)
\(c)\dfrac{0,8:\left(\dfrac{4}{5}.1,25\right)}{0,64-\dfrac{1}{25}}+\dfrac{\left(1,08-\dfrac{2}{25}\right):\dfrac{4}{7}}{\left(6\dfrac{5}{9}-3\dfrac{1}{4}\right).2\dfrac{2}{17}}+\left(1,2.0,5\right):\dfrac{4}{5}\)
\(d)158.[\dfrac{12-\dfrac{12}{7}-\dfrac{12}{289}-\dfrac{12}{85}}{4-\dfrac{4}{7}-\dfrac{4}{289}-\dfrac{4}{85}}:\dfrac{5+\dfrac{5}{13}+\dfrac{5}{169}+\dfrac{5}{91}}{6+\dfrac{6}{13}+\dfrac{6}{169}+\dfrac{6}{91}}].\dfrac{5050505050}{7171717171}\)
Bài 2:
a)Tính: \(M=(\dfrac{\dfrac{2}{5}-\dfrac{2}{9}+\dfrac{2}{11}}{\dfrac{7}{5}-\dfrac{7}{9}+\dfrac{7}{11}}-\dfrac{\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}}{1\dfrac{1}{6}-\dfrac{7}{8}+\dfrac{7}{10}}):\dfrac{2021}{2022}\)
b) Chứng tỏ \(M=75:\left(4^{2021}+4^{2020}+...+4^2+4+1\right)+25⋮10\)
c) Tìm \(x,y,z\inℕ^∗\)nhỏ nhất thỏa mãn 18\(x\)=24\(y\)=36\(z\)
Bài 3:
a) Tìm \(P\) là số nguyên tố để \(P+6,P+12,P+18,P+24\) là số nguyên tố
b)Tìm sô tự nhiên n thỏa mãn: \(2.2^2+3.2^3+...+4.2^4+...+n.2^n=2^{n+11}\)
Giúp mik nhe các bạn ☺
Bài 2:
a: \(M=\left(\dfrac{\dfrac{2}{5}-\dfrac{2}{9}+\dfrac{2}{11}}{\dfrac{7}{5}-\dfrac{7}{9}+\dfrac{7}{11}}-\dfrac{\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}}{1\dfrac{1}{6}-\dfrac{7}{8}+\dfrac{7}{10}}\right):\dfrac{2021}{2022}\)
\(=\left(\dfrac{2\left(\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{11}\right)}{7\left(\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{11}\right)}-\dfrac{\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}}{\dfrac{7}{6}-\dfrac{7}{8}+\dfrac{7}{10}}\right)\cdot\dfrac{2022}{2021}\)
\(=\left(\dfrac{2\left(\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{11}\right)}{7\left(\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{11}\right)}-\dfrac{\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}}{\dfrac{7}{2}\left(\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}\right)}\right)\cdot\dfrac{2022}{2021}\)
\(=\left(\dfrac{2}{7}-\dfrac{2}{7}\right)\cdot\dfrac{2022}{2021}\)
=0
b: Đặt \(N=4^{2021}+4^{2020}+...+4^2+4+1\)
=>M=75N+25
\(4N=4^{2022}+4^{2021}+...+4^3+4^2+4\)
=>\(4N-N=4^{2022}+4^{2021}+...+4^3+4^2+4-4^{2021}-4^{2020}-...-4^2-4-1\)
=>\(3N=4^{2022}-1\)
\(M=75N+25=25\left(3N+1\right)\)
\(=25\left(4^{2022}-1+1\right)\)
\(=25\cdot4^{2022}=100\cdot4^{2021}⋮10\)
c: 18x=24y=36z
=>\(\dfrac{18x}{72}=\dfrac{24y}{72}=\dfrac{36z}{72}\)
=>\(\dfrac{x}{4}=\dfrac{y}{3}=\dfrac{z}{2}\)
=>bộ số nguyên dương (x;y;z) nhỏ nhất thỏa mãn là (4;3;2)
Bài 3:
a: TH1: P=5
P+6=11; P+12=5+12=17; P+18=5+18=23; P+24=24+5=29
=>NHận
TH2: P=5k+1
P+24=5k+24+1=5k+25=5(k+5) chia hết cho 5
=>Loại
TH3: P=5k+2
P+18=5k+2+18=5k+20=5(k+4) chia hết cho 5
=>Loại
TH3: P=5k+3
P+12=5k+3+12=5k+15=5(k+3) chia hết cho 5
=>Loại
TH4: P=5k+4
P+6=5k+4+6=5k+10=5(k+2) chia hết cho 5
=>Loại
Vậy: P=5