解方程、不定方程3題

1)6x + 7y + 11z = 9893
設 6x + 7y = k
一組解為 x = - k , y = k
故通解為 x = - k - 7m , y = k + 6m而 k + 11z = 9893
一組解為 k = 4 , z = 899
故通解為 k = 4 - 11n , z = 899 + n綜上原不定方程通解為
x = - k - 7m = 11n - 4 - 7m
y = k + 6m = 4 - 11n + 6m
z = 899 + n
(m , n 為整數)
正整數解 :x = 11n - 4 - 7m > 0
y = 4 - 11n + 6m > 0
z = 899 + n > 011n - 4 - 7m = 0 交 4 - 11n + 6m = 0 於 A = (0 , 4/11)
4 - 11n + 6m = 0 交 899 + n = 0 於 B = (- 9893/6 , - 899)
899 + n = 0 交 11n - 4 - 7m = 0 於 C = (- 9893/7 , - 899)△ABC 內(不含邊界)的每一個整點與原不定方程的每一組正整數解一一對應。

2)6x + 7y + 11z = 9893 -------- ①
9x + 5y + 10z = 9889 -------- ②
由題 1) , ①式通解為 :
x = 11n - 4 - 7m
y = 4 - 11n + 6m
z = 899 + n代入② ,
9(11n - 4 - 7m) + 5(4 - 11n + 6m) + 10(899 + n) = 9889
18n - 11m = 305
一組解為 n = 20 , m = 5
通解為 n = 20 + 11t , m = 5 + 18t故原不定方程組通解為 :
x = 11n - 4 - 7m = 11(20 + 11t) - 4 - 7(5 + 18t) = 181 - 5t
y = 4 - 11n + 6m = 4 - 11(20 + 11t) + 6(5 + 18t) = - 13t - 186
z = 899 + n = 899 + 20 + 11t = 919 + 11t
正整數解 :x = 181 - 5t > 0
==>
t ≤ 36y = - 13t - 186 > 0
==>
t ≤ - 15z = 919 + 11t > 0
==>
t ≥ - 83 綜上, - 83 ≤ t ≤ - 15
代入此範圍之整數 t 值得原不定方程組共 - 15 - (- 83) + 1 = 69 組正整數解。

3)6x + 7y + 11z = 9893 -------- ①
9x + 5y + 10z = 9889 -------- ②
11x + 6y + 7z = 9884 -------- ③
由題 2) , ①,②之通解為
x = 181 - 5t
y = - 13t - 186
z = 919 + 11t代入③ ,
11(181 - 5t) + 6(- 13t - 186) + 7(919 + 11t) = 9884
t = - 46原方程組之解為
x = 181 - 5(-46) = 411
y = - 13(-46) - 186 = 412
z = 919 + 11(-46) = 413

For part (3), using Gaussian elimination, x = 411, y = 412 and z = 413.