>>13592443Adding to what
>>13592461 said
We notice that the last digit of the number we rise to a power is the only thing that matter for the last digit of the result.
In particular using the binomial theorem:
Because of this the only thing that we care about is the last digits of for .
And in those you can easily notice a pattern. Take for example 2^n. As your picture suggest there is a repetition. After we reach last digit equal to 2 again (in 2^5= 32) we stop because we know that we only need to consider the last digit and if we continue doing it we are going to end up doing the same stuff.
So basically you don't get zeroes because there are not zeroes in the 1,2,3,4,5,6,7,8,9 last digit power cycles. More precisely because for to hold we would need and so which only happens for . (so for numbers ending in 0)
