About Mathematics

5 pirates of different ages have a treasure of 100 gold coins. On their ship, they decide to split the coins using this scheme: The oldest pirate proposes how to share the coins, and ALL pirates (including the oldest) vote for or against it. If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain. As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard. Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?

I will provide two methods for this. Method 1  (Easy way) : USE A CALCULATOR. You will get the answer in a matter of seconds like I got. Clearly, the answer is 072. Method 2  (Slight harder way) : Here, I want to find the answer without using the calculator. Let's try it. Note :- I will be making use of  Congruence Modulo  and  Euler's Theorem , so these are the prerequisites. Another way to put the question is  “Find the remainder when  2 2017 2 2017  is divided by 1000”. First of all we factorise 1000 as: 1000 = 2 3 × 5 3 1000 = 2 3 × 5 3 Next, we find the remainder by  2 3 2 3  and  5 3 5 3 seperately. It's obvious that  2 3 2 3  divides  2 2017 2 2017 . Hence, 2 2017 ≡ 0 ( m o d 8 ) 2 2017 ≡ 0 ( m o d 8 ) Now to find the remainder by 125 (or 5^3), we use Euler's Theorem. Euler's theorem is applicable in this case since  g c d ( 2 , 125 ) = 1 g c d ( 2 , 125 ) = 1 . Φ ( 125 ) = ...