7n+19/2n+1为整数,则n=?(7n+19)/(2n+1)，n≠0

Question

7n+19/2n+1为整数,则n=?
(7n+19)/(2n+1)，n≠0

Answer 1

有括号吗?

如果没有括号的话,只要n是偶数即可.如：

n=2：7*2+(19/2)*2+1=14+19+1=34

n=4：7*4+(19/2)*4+1=28+38+1=67

.

如果有括号：

(7n+19)/(2n+1)

=(6n+3+n+16)/(2n+1)

=(6n+3)/(2n+1)+(n+16)/(2n+1)

=3+(n+16)/(2n+1)      只要(n+16)/(2n+1)是整数即可.

令(n+16)/(2n+1)=a

n+16=a(2n+1)

n+1/2+31/2=a(2n+1)

a(2n+1)-(2n+1)/2=31/2

(2n+1)(a-1/2)=31/2

31/2=1*(31/2)=31*(1/2)

1.   2n+1=1   n=0      不符合题意,舍去；

2.  2n+1=31   n=15     代入得

(7*15+19)/(2*15+1)

=(105+19)/(31)

=124/31

=4

所以n=15