Number Theory (Linear 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

Suppose they have x, y and z and p in the purse

1st man

x + p + 2 (y + z)

2nd man

y + p = 3 (x + z)

3rd man

z + p = 5 (x + y)

This gives 3 equations in 4 unknowns (so no unique solution).

You can get x, y,z in terms of p.

[ from 1st equation write x = ....., then substitute for x in the 2nd and 3rd equation and solve for y,z in terms of p. Then find x in terms of p ]

I got : x=p/15, y = p/5, z = p/3