Question

[Danh Ha Anh]

Tìm x nguyên để các phân số sau là số nguyên:

a)\(\frac{-3}{x-1}\)

b)\(\frac{-4}{2x-1}\)

c)\(\frac{3x+7}{x-1}\)

d)\(\frac{4x-1}{3-x}\)

Answer 1

[Dương Lam Hàng]

a) Để \(\frac{-3}{x-1}\in Z\) \(\Leftrightarrow-3⋮\left(x-1\right)\)

   \(\Rightarrow x-1\inƯ\left(-3\right)=\left\{-1;1;-3;3\right\}\)

   \(\Rightarrow x=\left\{2;0;4;-2\right\}\)

b) Để \(\frac{-4}{2x-1}\in Z\Leftrightarrow-4⋮\left(2x-1\right)\)

\(\Rightarrow2x-1\inƯ\left(-4\right)=\left\{-1;1;-2;2;-4;4\right\}\)

\(\Rightarrow2x=\left\{0;2;-1;3;-3;5\right\}\)

\(\Rightarrow x=\left\{0;1;\frac{-1}{2};\frac{3}{2};\frac{-3}{2};\frac{5}{2}\right\}\)

Mà \(x\in Z\) \(\Rightarrow x=\left\{0;2\right\}\)

c) \(\frac{3x+7}{x-1}=\frac{3\left(x-1\right)+10}{x-1}\)

Vì \(3\left(x-1\right)⋮\left(x-1\right)\Rightarrow10⋮\left(x-1\right)\)

\(\Rightarrow x-1\inƯ\left(10\right)=\left\{1;-1;2;-2;5;-5;10;-10\right\}\)

\(\Rightarrow x=\left\{2;0;3;-1;6;-4;11;-9\right\}\)

d) Tương tự