Prime Power Math Puzzle

Source: IBM Ponder This - June 2012 - Sent to me by Aashay Harlalka (Final Year Student, CSE, IITB)


Two players starts with the number N and play in turns. In each turn the player chooses a prime power p^m > 1, which divides N and updates N to be N/(p^m). The player who sets N to be 1 - wins.
What are all the winning moves from the initial state of N=1,506,009,006,001,500,000,000 ?

We are looking for all the moves the first player can make which, assuming he plays correctly, can guarantee him winning the game.

Update (24 June 2014):
Solution: Posted by Gowtham Kumar (PhD Student, Stanford, IITM Alumnus) and me in comments!


  1. This is a Nim game. Google Nim game for answer.

  2. First we factor N= 1,506,009,006,001,500,000,000.
    So the factorization of N is 2^8 * 3^1 * 5^9 * 7^4 * 11^4 * 13^4.

    Then we note that the game is a actually the classic game of Nim in disguise.

    The initial state is (8,1,9,4,4,4)

    Nim Game on Wikipedia:


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