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

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?

Answer 1

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.

Answer 2

Thank you so much!

Answer 3

To find the minimum number of original coconuts, we need to work backwards from the given information.

Let's assume x is the number of original coconuts. According to the information given, the number of coconuts left after each man took their share can be represented as:

First man: (4/5)x
Second man: (4/5)(4/5)x = (4/5)^2x
Third man: (4/5)^3x
Fourth man: (4/5)^4x
Fifth man: (4/5)^5x

We are also told that the remaining coconuts after all the men took their share were then evenly divided among themselves and the leftover coconut was given to the monkey. This implies that the number of coconuts left after each man took their share must be divisible by 5.

Using this condition, we can set up an equation:

(4/5)^5x = 5y

Simplifying this equation, we get:

(1024/3125)x = 5y

To find the minimum number of original coconuts, we need to find the smallest possible values for x and y that satisfy this equation.

The smallest values that satisfy this equation are x = 3125 (5^5) and y = 1024 (2^10).

Therefore, the minimum number of original coconuts is 3125.

Note: This problem involves a mathematical concept known as inverse operations or working backwards to find the original quantity. In this case, we worked backwards from the given information to find the minimum number of original coconuts.