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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

Dwarf Longnose was helped by some squirrels, some hamsters and some pigeons. If the number of squirrels and the number of hamsters are in a ratio of 3:4 and the number of hamsters is in a ratio of 7:5 with the number of pigeons, what is the smallest possible number of animals that helped Longnose?

(characters from Wilhelm Hauff's "Dwarf Longnose")

From the first ratio, there are 3x squirrels and 4x hamsters.

Similarly, from the second ratio there are 7y hamsters and 5y pigeons.

However, the number of hamsters is one and the same:

equation

Since the number of hamsters is necessarily an integer number, the equation is Diophantine. It can be solved by identifying the two prime factorizations:

equation

Since 4 and 7 are coprime, the smallest number for which the equation is satisfied is:

equation

Therefore,

equation

equation

The animal counts are:

equation

equation

equation

and the total number of animal helpers is:

equation