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a) a + b = 270 và ƯCLN(a, b) = 45.
b) a.b = 300 và ƯCLN(a, b) = 5.
c) a.b = 2700 và BCNN(a, b) = 900.
a: ƯCLN(a,b)=45
=>\(\left\{{}\begin{matrix}a=45k\\b=45c\end{matrix}\right.\)
Ta có: a+b=270
=>45k+45c=270
=>45(k+c)=270
=>k+c=6
=>\(\left(k;c\right)\in\left\{\left(1;5\right);\left(2;4\right);\left(3;3\right);\left(4;2\right);\left(5;1\right)\right\}\)
=>\(\left(a,b\right)\in\left\{\left(45;225\right);\left(90;180\right);\left(135;135\right);\left(180;90\right);\left(225;45\right)\right\}\)
mà ƯCLN(a,b)=45
nên \(\left(a,b\right)\in\left\{\left(45;225\right);\left(225;45\right)\right\}\)
b: \(ƯCLN\left(a,b\right)=5\)
=>\(\left\{{}\begin{matrix}a=5k\\b=5c\end{matrix}\right.\)
\(5k\cdot5c=300\)
=>\(25\cdot k\cdot c=300\)
=>\(k\cdot c=\dfrac{300}{25}=12\)
=>\(\left(k;c\right)\in\left\{\left(1;12\right);\left(12;1\right);\left(2;6\right);\left(6;2\right);\left(3;4\right);\left(4;3\right)\right\}\)
=>\(\left(a,b\right)\in\left\{\left(5;60\right);\left(60;5\right);\left(10;30\right);\left(30;10\right);\left(15;20\right);\left(20;15\right)\right\}\)
mà ƯCLN(a,b)=5
nên \(\left(a,b\right)\in\left\{\left(5;60\right);\left(60;5\right);\left(15;20\right);\left(20;15\right)\right\}\)
a/ \(ƯCLN\left(a,b\right)=16\) \(\Leftrightarrow\left\{{}\begin{matrix}a=16.a_1\\b=16.b_1\end{matrix}\right.\) \(\left(a_1;b_1\right)=1\) (1)
Thay \(\left(1\right)\) vào \(a+b=128\)
\(\Leftrightarrow a+b=16.a_1+16.b_1=128\)
\(\Leftrightarrow16\left(a_1+b_1\right)=128\)
\(\Leftrightarrow a_1+b_1=8\)
Mà \(\left(a_1;b_1\right)=1\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}a_1=1\\b_1=7\end{matrix}\right.\\\left\{{}\begin{matrix}a_1=3\\b_1=5\end{matrix}\right.\\\left\{{}\begin{matrix}a_1=7\\b_1=1\end{matrix}\right.\\\left\{{}\begin{matrix}a_1=5\\b_1=3\end{matrix}\right.\end{matrix}\right.\)
Mà \(\left\{{}\begin{matrix}a=16.a_1\\b=16.b_1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}a=16\\b=112\end{matrix}\right.\\\left\{{}\begin{matrix}a=48\\b=80\end{matrix}\right.\\\left\{{}\begin{matrix}a=112\\b=16\end{matrix}\right.\\\left\{{}\begin{matrix}a=80\\b=48\end{matrix}\right.\end{matrix}\right.\)
Vậy ..
cho mình hỏi a,b là số chung à