In words, show that for all n there exists integers k and j such that the number of divisors of j plus the number of numbers coprime to and less than k is n.

So, I would note that so taking , .

However, this is only for the natural numbers and zero...not the whole integers. You can't get a negative number of divisors!

It is also a somewhat boring result - why the Euler totient function if we can just forget about it?...

Have you perhaps missed out something from the question?