Five men and a leopard

Question:

Five men and a leopard were shipwrecked on a desert island, and they spent the first day gathering coconuts for food, piled the coconuts all up together, and then went to sleep for the night.

But when they all were asleep one man woke up,and he divided the coconuts into five piles. He had one coconut left over, and he gave that to the leopard, and he hid his pile and put the rest all back together. By and by the next man woke up and did the same thing. And he had one left over, and he gave it to the leopard.

And all five of the men did the same thing, one after the other; each one taking one fifth of the coconuts in the pile when he woke up, and each one having one left over for the leopard. And in the morning they divided what coconuts were left, and they came out in five equal shares. Of course each one must have known there were coconuts missing, but each one was guilty as the others, so they didn’t say anything.
How many coconuts were there in the beginning (smallest number)?

.

.

C

A

P

T

A

I

N

I

N

T

E

R

V

I

E

W

.

.

Answer:  3121
Solution: possibly it can be cut short. 4/5(4/5(4/5(4/5(4/5(n-1)-1)-1)-1) – 1) – 1 = 5m. We need to find n. ==> 5/4(5/4(…(5m+1)…)+1)+1 = n ==> ((5^6)*m +5^5+5 ^6 -4^6)/4^5 = integer (i.e n) ==> We need ‘m’ s.t 4^5|(5m+6) ==> 1024*(?)=5m+6 ==> ? has to be 4 to get 6 in units place ==> so we get 5m+6 and hence n. The answer turns out to 3121.

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