Five men collected several coconuts from a deserted island, and they decided to divide thecoconuts the next day. During the night, one of them decided to separate his part so he divided the total in 5 groups and give the left-over coconut to a nearby monkey, the man went to sleep. Next, another man did the same with the remainders coconuts and give the left-over coconut to the monkey. During the night, every man did the same. Next morning the men divide the remainder coconuts and give the left-over coconut to the monkey. Question: What is the minimum number of original coconuts?

Question

mathhelper · Accepted Answer

before 5th division, number of coconuts = 5k+1
before 4th division, number of coconuts = 5(5k+1) + 1 = 25k + 6
before 3rd division, number of coconuts = 5(25k + 6) + 1 = 125k + 31
before 2nd division, number of coconuts = 5(125k + 31) + 1 = 625k + 156
before 1st division, number of coconuts = 5(625k + 156) + 1
= 3125k + 781

let k = 0 , number of coconuts is 781

check:
after 1st split: 781 ÷ 5 = 156, remainder 1
after 2nd split: 156 ÷ 5 = 31, remainder of 1
after 3rd split: 31 ÷ 5 = 6, remainder of 1
ater 4th split: 6 ÷ 5 = 1, remainder of 1
after 5th split: 1 ÷ 5 = 0, remainder of 1

mathematically this works, since before the last split, there would be
1 coconut left, implying none for each of the 5 men, but one for the monkey.

if the men are to receive any coconuts at all, let k = 1
number would be 3125 + 781 = 3906 , which would leave 6 at the end with 1 left over for the monkey.

Me · Answer

Thank you so much!