Toán Lớp 8: nếu ( a+b+c ) chia hết 30 thì (a^5+b^5+c^5 ) chia hết 30

Question

Toán Lớp 8:  nếu ( a+b+c ) chia hết 30 thì (a^5+b^5+c^5 ) chia hết 30

quynhmaithu · Answer

Giải đáp:      Ta c&oacute; :     a^5&minus;a   =a.(a^4&minus;1)   =a.(a^2&minus;1).(a^2+1)   =a.(a&minus;1).(a+1).(a^2&minus;4+5)   =a.(a&minus;1).(a+1).(a^2&minus;4)+5a.(a&minus;1).(a+1)   =(a&minus;2).(a&minus;1)a(a+1)(a+2)+5a(a&minus;1)(a+1)   Ta thấy : (a&minus;2)(a&minus;1)a(a+1)(a+2) l&agrave; t&iacute;ch 5 số tự nhi&ecirc;n li&ecirc;n tiếp   -> (a&minus;2)(a&minus;1)a(a+1)(a+2)⋮5 v&agrave; cũng ⋮6   -> (a&minus;2)(a&minus;1)a(a+1)(a+2)⋮ 5.6 = 30   m&agrave;: 5 ⋮5    a(a - 1)(a + 1) l&agrave; 3 số tự nhi&ecirc;n li&ecirc;n tiếp   -> a(a - 1)(a + 1)⋮6   -> 5a.(a&minus;1).(a+1) ⋮ 5.6 = 30   -> (a&minus;2).(a&minus;1)a(a+1)(a+2)+5a(a&minus;1)(a+1) ⋮  30   -> a^5&minus;a ⋮  30   Cmtt -> b^5&minus;b v&agrave; c^5&minus;c⋮  30   -> a^5+b^5+c^5&minus;(a+b+c) ⋮  30   m&agrave;: a+b+c ⋮  30   -> a^5+b^5+c^5 ⋮  30    ∘ dariana