Cardan was a Renaissance mathematician. The following problems appeared in Carda

Question

Cardan was a Renaissance mathematician. The following problems appeared in Cardan's Ars Magna: There were two leaders, each of whom divided 48 aurei (gold coin of ancient Rome) among his soldiers. One of these have two more soldiers than the other. The one who two soldiers fewer had four aurei more for each soldier. Find how many soldiers each had.

Explanation / Answer

sol: Let leaders have x and x+2 soldiers let the leader with x soldiers give y aurei each; the leader with x+2 soldiers gave y-4 each Then xy=48 and (x+2)(y-4)=48 xy+2y-4x-8=48 As xy=48 then 2y-4x-8=0 or y=2x+4 Now use that xy=48 so x(2x+4)=48 => x^2+2x-24=0 => (x+6)(x-4)=0 so x=4 or -6 As number of soldiers is positive (!) must have x=4 Thus one leader has 4 soldiers with 12 aurei each The other has 6 soldiers with 8 each

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