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Math Help - my first problem

  1. #1
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    my first problem

    let a1,a2,...........,an be n nubers such that each ai is either 1 or -1.if a1a2a3a4+a2a3a4a5+........+ana1a2a3=0,
    prove that 4 divides n


    pls give me a solution
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  2. #2
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    Quote Originally Posted by arjun sengupta View Post
    let a1,a2,...........,an be n nubers such that each ai is either 1 or -1.if a1a2a3a4+a2a3a4a5+........+ana1a2a3=0,
    prove that 4 divides n


    pls give me a solution
    Any of the ai's come in exactly 4 terms out of (a1a2a3a4,a2a3a4a5,...,ana1a2a3).
    Let the sum be S0. Now just change the sign of any one of the ai. Let the new sum be S1. It is easy to see that 4|(S1-S0)

    Now initially if all ai's =1; sum = n
    Final sum when some ai's are -1 = 0
    thus 4|(n-0) => 4|n
    Last edited by aman_cc; November 19th 2009 at 06:58 AM.
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