Arabian Nights!

Number Theory Level 5

An old man willed that upon his death, his three sons would receive the \(u^{th}\), \(v^{th}\) and \(w^{th}\) parts of his herd of camels respectively. He had \(N\) camels in the herd when he died, where \(N+1\) is a common multiple of \(u,v\) and \(w\).

Since the three sons couldn't divide \(N\) exactly into \(u,v\) or \(w\) parts, they approached Calvin for help. Calvin rode over on his own camel, which he added to the herd. The herd was then divided up according to the old man's wishes. Calvin then took back the one camel that remained, which was, of course, his own.

How many un-ordered pairs of solutions \((u, \ v, \ w, \ N)\) exists satisfying the above conditions?


  • Solve the same problem to find all the solutions if the old man had four sons with similar conditions.

  • Let there be \(k\) sons. Find an upper bound \(f(k)\) on \(N\) for the problem to have a solution.

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