Recall Euler's theorem: if 
, then
where 
is Euler's totient function.
We have 
- in fact, 
for any 
since 
and 
share no common divisors - as well as 
.
Now,
where the 
are positive integer coefficients from the binomial expansion. By Euler's theorem,
so that
69,420,911,80085 and that’s all
The answer is one in six because there's six sides and only one four on the die
