Find the last three digits of the above number.
Let and be positive integers with such that for all integers
Find the smallest possible value of
Clarification: denotes the factorial notation. For example, .
The sequence above can be recursively defined by
Given a positive integer , a new sequence is created by dividing each term by and writing down the remainder of the division. We denote this new sequence by .
For a given , it is known that the terms of eventually become constant. Let denote the index at which the constant values begin, i.e. is the smallest number for which .
Find .
How many integers are there such that and end with the same digit?
An automorphic number is defined as a positive integer such that the trailing digits of , where is a positive integer, is itself for all .
Let us define an almost automorphic number as a number where only appears as the trailing digits of for all odd . How many almost automorphic numbers are less than 1000?