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
Answer
Find out the how long did they rest in 8 hours.
To prove
As given
contestants in a dance-a-thon rest for the same amount of time every hour.
A couple rests for 25 minutes in 5 hours .
As
25 mintues = 5 hours
1 hours = 5 mintues
Thus in 1hours a the dance -a-thon rest for 5 mintues.
Now for 8 hours.
8 hours = 8 × 5 mintues
= 40 mintues
Therefore in 8 hours a dance-a-thon rest for 40 mintues.