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Math Help - number theory: proof required!

  1. #1
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    number theory: proof required!

    hi

    I've been given the following problem:

    Prove that the last digit of the product of two consecutive integers is 0, 2 or 6.

    So far, I can prove it's not an odd number:

    n and (n+1) are two consecutive integers - one is odd, and the other even, so, let n=2x and (N+1) = 2x + 1

    so n(n+1) = 2x(2x+1) and so an even number.

    and write out a table:

    0x1 = 0
    1x2 = 2
    2x3 = 6
    3x4 = 12 (last digit is 2)
    4x5 = 20 (last digit is 0)
    5x6 = 30 (last digit is 0)
    6x7 = 42 (last digit is 2)
    7x8 = 56 (last digit is 6)
    8x9 = 72 (last digit is 2)

    and I can justify to myself that clearly when you multiply any two consecutive integers, the last digit will be a 0, 2 or 6; but I can't prove this (to either myself or my uni lecturer!)

    Any nudges in the right direction? please?!
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  2. #2
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    after two days of banging head against desk; I've come up with an unelegant solution
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  3. #3
    MHF Contributor chiph588@'s Avatar
    Joined
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    Champaign, Illinois
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    Notice that the last digit of two multiplied numbers depends only on the last digit of each number being multiplied.

    Thus you just need to verify it for  n=0 through  n=9 .
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