stone hull
Hi! I saw the solution to this problem but I can't solve it using modular arithmetic. The solution is as follows:
(a^9)-a = a((a^8)-1)
(a^9)-a = a(((a^4)^2)-(1^2))
(a^9)-a = a((a^4)+1)((a^4)-1)
(a^9)-a = a((a^4)+1)((a^2)+1)((a^2)-1)
(a^9)-a = a((a^4)+1)((a^2)+1)((a+1)(a-1)
(a^9)-a = ((a^4)+1)((a^2)+1) (a+1) a(a-1)
and since a(a-1) is divisible by 2 and (a+1)a(a-1) is divisible by 3 (because one of the terms will necessarily be a multiple of 3), (a^9)-a is divisible by 6.
I am really bad with congruences and modular arithmetic, so I saw that as a good practice exercise. I tried using fermat's little theorem too, but it doesn't lead nowhere and I get answers that don't make sense. I suspect that the solution is easy, but I need to understand that first. Could someone here enlighten me please?
sleek adder