[理工] [離散] 整係數方程式求解

[movo11 (Larry)]

 1.True or false: the equation 12x + 9y = 16 has an integral equation.

 2.Which of the following linear equations has solution in integers?
   Please show the reason.(1)28x+16y=6 (2)51x+18y=12(3)21x+35y=45

 3.Show that x^3 + 5y^3 = 25z^3 has no solutions in positive integers
   x,y,and z.

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◆ From: 123.192.157.110
→ sting47:1.F 2.(2)
推 ddczx:1.F 2.(2) 3.因x,y,z若有公因數,同除消去後亦可滿足原方程,
→ ddczx:令(x,y,z)=(a,b,c)且gcd(a,b,c)=1為一解
→ ddczx:則 a^3 + 5b^3 = 25c^3 -> a為5的倍數 -> a=5t for some t
→ ddczx:原式=125t^3+ 5b^3 = 25c^3 -> 25t^3+ b^3 = 5c^3
→ ddczx:繼續剛剛步驟做下去可得b,c均為5之倍數,與gcd(a,b,c)=1不合