Question

[mạnh dũng nguyễn]

a) a + b = 270 và ƯCLN(a, b) = 45.
b) a.b = 300 và UCLN(a, b) = 5.
c) a.b = 2700 và BCNN(a, b) = 900.

Answer 1

[Nguyễn Lê Phước Thịnh]

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 \(\left(a,b\right)\in\left\{\left(5;60\right);\left(60;5\right);\left(15;20\right);\left(20;15\right)\right\}\)