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the 9 bull elephants and three female elephants have learned how to share their peanuts equally. First, every bull gives an equal amount to every female. Then every female gives a different, equal amount to every bull. Each elephant then possessed the same amount of peanuts - the smallest possible number. How many peanuts does each elephant have?   

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Stephen K. | Physics PhD experienced in teaching undergraduatesPhysics PhD experienced in teaching unde...
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The smallest # of peanuts must be a number that can be divided equally by 9 (since each bull gives an equal # to each female) and be equally divided by 12 (since each elephant must end up with an equal # of peanuts.
The smallest number which is a multiple of both 9 and 12 is 9 x 12 = 108.
Then each bull gives 12/3 = 4 peanuts to each female.
Each female then ends up with 9 x 4 = 36 peanuts.
Since the females make up 1/4 of the total population of 12 elephants, they keep 1/4 of the total number of peanuts and divide the other 3/4 equally between the males, i.e. each female keeps 9 peanuts and returns 27.
27/9 = 3 so each female gives each male 3 peanuts so that the total # of peanuts received by each male is 9.
Each elephant then has 9 peanuts.