Results 1 to 3 of 3

Math Help - Factorials & Perfect Squares

  1. #1
    Junior Member star_tenshi's Avatar
    Joined
    Jan 2009
    Posts
    32

    Factorials & Perfect Squares

    I've been racking my brain out for the last two days on this problem, but I don't even have the slightest clue on how to approach this problem. Could anybody point me in the right direction please?

    Q: Find the values of n \geq 1 for which  1! + 2! + ... + n! is a perfect square. (Hint: Determine the possibilities for the units digit of a perfect square.)

    Using the hint:
     0^2 = 0
    1^2 = 1
    2^2 = 4
    3^2 = 9
    4^2 = 16
    5^2 = 25
    6^2 = 36
    7^2 = 49
    8^2 = 64
    9^2 = 81
    10^2 = 100

    This leads to a pattern: 0 1 4 9 6 5 6 9 4 1 0

    Then I'm just lost....
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,245
    Thanks
    1
    The unit digit of 1! is 1.

    The unit digit of 2! is 2.

    The unit digit of 3! is 6.

    The unit digit of 4! is 4.

    The unit digit of n!, \ n\geq 5 is 0.

    Now, we have:
    1!=1=1^2

    1!+2!+3!=1+2+6=9=3^2

    For n\geq 4 the unit digit of 1!+2!+3!+\ldots+n! is 3 and it can't be a perfect square.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member star_tenshi's Avatar
    Joined
    Jan 2009
    Posts
    32
    Thanks a lot red_dog.

    After your hint, I completed the mathematical proof and finished off the problem.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Perfect Squares
    Posted in the Algebra Forum
    Replies: 3
    Last Post: September 6th 2011, 03:46 PM
  2. Perfect Squares
    Posted in the Math Puzzles Forum
    Replies: 1
    Last Post: July 19th 2010, 12:36 AM
  3. perfect squares
    Posted in the Algebra Forum
    Replies: 3
    Last Post: December 13th 2009, 03:20 AM
  4. Perfect Square of a Product of Factorials Proof
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: December 10th 2007, 09:29 PM
  5. Perfect Squares
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: March 18th 2007, 05:05 PM

Search Tags


/mathhelpforum @mathhelpforum