[Cryptography] updating a counter

[John Denker]

On 05/19/2014 12:59 PM, Ben Laurie wrote:
>> > Each subcounter gets updated every time.  I arrange for
>> > the periods to be relatively prime, so that the period
>> > of the collection is the product of the periods of the
>> > individual subcounters.

> Hmm. That's surely not sufficient to ensure it doesn't repeat values.
> Which matters.

Actually it *is* sufficient to ensure that the overall
counter does not "repeat values" until its overall period
expires.  Consider a collection of four subcounters, with 
periods
  2^32
  2^32 - 1
  2^31 - 1
  2^29 - 1

The overall period is very nearly 2^124.  Proof:  Immediate
consequence of the fundamental theorem of arithmetic.
  http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic
plus the fact that the block cipher is invertible.

On a verrry fast machine that will "repeat" once in 10^9
times the age of the universe.  So yeah, it "repeats".
Please explain in more detail why that matters.

If you don't believe the theorem, please provide a counterexample.