# An application of quadratic residues

A certain Count had 5 daughters and 11 peasants. Upon his death he willed that the square shaped county be divided into equal square plots such that each daughter gets one plot and the remaining plots are divided evenly among the peasants. Is there a solution?

Yes: Divide the county into 49 plots. 5 plots go to the daughters and the remaining 44 can be divided evenly.

Is there a solution if he had 2 daughters and 11 peasants? The answer turns out to be no and it lies in the fact that
$x^2 \equiv 2 \pmod{11}$ has no solution.

The story is taken from Pommersheim et al.

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### 3 Responses to An application of quadratic residues

1. magnus says:

5 11=4*4=16. Why use 49?

2. tpc says:

Yes, 16 also works. Thanks for pointing it out.

3. lakesh kumar ravi says:

16 can’t work because 5 plots go to the daughter but the remaining 11 plots can’t be divided onto 11 peasant evenly..