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The monkey and the coconuts

The monkey and the coconuts

Five men are shipwrecked on a desert island where a lonely monkey lives. The men spend the entire day collecting coconuts for food and decide to distribute them among all the next day. At night one of them wakes up and distrustful, decides to separate his part. Divide the coconuts into five piles and take their share and as a coconut it is given to the monkey. Shortly after a second castaway wakes up and does the same. When dividing the coconuts into five piles, a coconut is left over and also given to the monkey. One after another all shipwrecked do the same. The next day in the morning none of the shipwrecked confesses and the remaining coconuts are divided into five piles without any on top.

How many coconuts had they collected exactly if we know that every hour a castaway collected between 50 and 55 coconuts and that they had been collecting coconuts for 12 hours?

Solution

If we represent the number of coconuts taken by each shipwreck and the total stock,
The following system of equations originates:

It is a system of diophantine equations in which the possible solutions must be whole numbers.

Where the result is:

So that the total number of coconuts collected was 3121 since according to the statement they collected between 50 × 5 × 12 = 3000 and 55 × 5 × 12 = 3300 coconuts and other solutions of the equation would be out of this range.