Prove that 2 of the n^k mod10 evaluated for 11 different values of k will be the same number. Or think how many possible different remainders can exist mod10?
I know I am supposed to PHP to figure this out, but I just can't do it.
Let n be a natural number.
Show that if e1, e2, ..., e11 are different positive integers, then there are two of them, call them ei, ej, such that (n^ei) - (n^ej) is divisible by 10. Any ideas? Thanks.