Question

Tìm n thuộc N để biểu thức sau là số tự nhiên

A= 2n+9/n+2 - 3n/n+2 + 5n+17/n+2

Answer 1

\(A=\dfrac{2n+9}{n+2}-\dfrac{3n}{n+2}+\dfrac{5n+17}{n+2}\)

\(A=\dfrac{2n+9-3n+5n+17}{n+2}\)

\(A=\dfrac{\left(2n-3n+5n\right)+9+17}{n+2}\)

\(A=\dfrac{4n+26}{n+2}\)

\(A=\dfrac{4n+8+18}{n+2}\)

\(A=\dfrac{4n+8}{n+2}+\dfrac{18}{n+2}\)

\(A=4+\dfrac{18}{n+2}\)

\(\Rightarrow18⋮\left(n+2\right)\Rightarrow n+2\inƯ\left(18\right)\)

n+2	1	2	3	6	9	18
n	-1	0	1	4	7	16

Vì \(n\in N\Rightarrow n\in\left\{0;1;4;7;16\right\}\)