Number theory (Diophantine Equation )

Question

Diophantine Equation

Three men came across a purse and picked it up. The first man, after looking at its contents, said: "If I get that purse, I'll have twice as much as your combined money", the second, after having done the same, said: "If I keep the purse, I'll have thrice as much as both of you have". The third, said: "If I'll get that purse, I'll have five times as much as your combined money. How much money was in the purse? How much money does each have?

Answer 1

Let M₁, M₂, M₃ = amount of money held by each man, and P equal amount of money in the purse.

The three conditions are:

M₁ + P = 2(M₂ + M₃)

M₂ + P = 3(M₁ + M₃)

M₃ + P = 5(M₁ + M₂)

Re-arranging and putting into matrix form Am = p (A is a 3x3, m and p are 3 x 1):

A = [-1 2 2, 3 -1 3, 5 5 -1][M₁ M₂ M₃]^T = [P P P]^T

Then M = A^-1P = [(1/15)P (3/15)P (5/15)P]

Setting P = 15 gives the smallest whole number solution: M₁ = 1, M₂ = 3, M₃ = 5, P = 15