Ta tìm số tự nhiên n để \(\frac{n+7}{n-2}\) rút gọn được
Gọi d là ước chung nguyên tố của n + 7 và n - 2
=> n+ 7 chia hết cho d
n - 2 chia hết cho d
=> (n+7) - (n- 2) chia hết cho d => 9 chia hết cho d
Mà d nguyên tố => d = 3
=> tìm n để n + 7 chia hết cho 3 và n - 2 chia hết cho 3
Do n + 7 = (n - 2) + 9 nên nếu n - 2 chia hết cho 3 thì n+ 7 sẽ chia hết cho 3
Vậy chỉ cần tìm n để n - 2 chia hết cho 3 => n - 2 = 3k (k \(\in\) N* vì n > 2) => n = 3k + 2
Với n = 3k + 2 (k \(\in\) N*) thì \(\frac{n+7}{n-2}\) rút gọn được
=> Với n \(\ne\) 3k + 2 (k \(\in\) N*) hay n là số chia hết cho 3 hoặc chia cho 3 dư 1 thì \(\frac{n+7}{n-2}\) tối giản
ta có :n+7/n-2 là p/s tối giản <=>uwcln(n+7,n-2)=1
gọi uwcln(n+7,n-2)=d
ta có n+7=n-2 +9
=>9 chia hết cho n-2
=>n-2 thuộc ư(9)={+-1;+-3;+-9}
=>n thuộc {3,1,5,-1,11,-7}
vì n>2 =>n={3,5,11}
vậy n thuộc {3,5,11}
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Để n+7/n-2 tối giản=>N+7 chia hết cho n-2
Ta có:N+7 chia hết cho n+2
=>(N+2)+5 chia hết cho n+2
=>5 chia hết cho n+2(Vì n+2 chi hết cho n+2=>5 chia hết cho n+2)
=>N+2 thuộc ƯC(5)
=>n+2 thuộc{1;-1;5;-5}
=>n thuộc{-1;-3;3;-7}
Vì n>2=>n thuộc{3}
Vậy n thuộc{3}
