2006 SMT/Algebra Problems/Problem 1
Problem
A finite sequence of positive integers for are defined so that and for . How many of these integers are divisible by ?
Solution
Notice that . Also, if is divisible by , then and is also divisible by . Therefore, all numbers of the form are divisible by .
Now, if we have a number of the form , this is equal to and is therefore more than a multiple of . Similarly, is more than a multiple of . Thus, the only divisible by are those in which is a multiple of .
The multiples of in are , and dividing all of these by we get . Therefore, there are multiples of .