a,b是正整数,证明30整除ab(a^4-b^4)

a^5-a
=a(a+1)(a-1)(aa+1)
=a(a+1)(a-1)(aa-4+5)
=a(a+1)(a-1)(aa-4)+5a(a+1)(a-1)
=(a-2)(a-1)a(a+1)(a+2) + 5(a-1)a(a+1)
已知连续5个数必有一个被5整除
已知连续2个数必有一个被2整除
已知连续3个数必有一个被3整除
(2*3*5)|(a-2)(a-1)a(a+1)(a+2)
(2*3)|(a-1)a(a+1)
=>
30|(a-2)(a-1)a(a+1)(a+2)
30|5(a-1)a(a+1)
=>
30|a^5-a
原式=a^5*b - ab^5
=a^5*b -ab -(ab^5 -ab)
=b(a^5 -a) - a(b^5 -b)
=>
30|b(a^5 -a)
30|a(b^5 -b)
=>
30|ab(a^4-b^4)