不等式和最小可取值

3x + 5y = 1, so y = (1 - 3x)/5
so S = x - y = x - (1 - 3x)/5 > 2007
5x - (1 - 3x) > 10035
5x - 1 + 3x > 10035
8x > 10036
x > 1254.5
so the smallest integer for x is 1255 or x must be an integer greater than 1255.... (1)
since y = (1 - 3x)/5 must also be an integer, that means ( 1 - 3x) must be a multiple of 5. So the last digit of x must be 7 or 2 ( example : 3 x 7 - 1 = 20, 3 x 2 - 1 = 5) .............(2)
To fulfill (1) and (2), the smallest no. of x is 1257, so y = [1 - 3(1257)]/5 = -754
so smallest value of S = 1257 - (-754) = 2011.

3x + 5y = 1
3x = 1 - 5y
x = (1 - 5y) / 3
x = ( (3 - 3y) - (2 - 2y) ) / 3
x = 1 - y - 2(1 - y) / 3

因 2 和 3 互質 , 故 1 - y 必為 3 的倍數。

令 1 - y = 3t , (t 為整數) , 則 y = 3t - 1 , 代回 :

3x + 5(3t - 1) = 1
x = 2 - 5t

x , y 通式為 :
x = 2 - 5t
{
y = 3t - 1

;S = x - y= (2 - 5t) - (3t - 1)= 3 - 8t
3 - 8t > 2007t < - 250.5當 t = - 251 S 的最小值 = 3 - 8(-251) = 2011

2011-09-16 23:41:06 補充：
Sorry! 前半段的1 - y 應是1 + y (打錯正負號) , 修改如下 :

3x + 5y = 1
3x = 1 - 5y
x = (1 - 5y) / 3
x = ( (3 - 3y) - (2 + 2y) ) / 3
x = 1 - y - 2(1 + y) / 3

因 2 和 3 互質 , 故 1 + y 必為 3 的倍數。

令 1 + y = 3t , (t 為整數) , 則 y = 3t - 1 , 代回 :

3x + 5(3t - 1) = 1
x = 2 - 5t

(下同)

2011-09-17 00:15:55 補充：
解法二 :

3x + 5y = 1

3x - 3y = 1 - 8y

x - y = (1 - 8y) / 3

;

(1 - 8y) / 3 > 2007

1 - 8y > 6021

y < - 752.5

整數 y < - 753

當 y = - 753 或 - 754 或 - 755 .......

S 最小值 = (1 - 8(- 753) ) / 3 = 2008 + 1/3 不合

S 最小值 = (1 - 8(- 754) ) / 3 = 2011 (答案)