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B Is there a lemma named for this?

  1. May 17, 2016 #1
    I'll call it the "Wheel Lug Lemma" for now.

    If there are a pair of integers p & q such that the Greatest Common Denominator is 1, and there is some number s that is product of p and an increasing whole number n, then the remainder of the division of s by q will cycle through all values of from 0 up to, but not including q until n is equal to q at which time the remainder is 0, with the cycle repeating again in the same order.

    The idea is if there is a wheel with p # of lug nuts, and the lug nuts are tightened in the order skipping q # of lug nuts to tighten the next nut, then eventually every lug nut will get tightened before encountering one that has already been tightened.

    For example, typically a wheel has q = 5 lug nuts, and it is recommended that they be tightened in a star pattern, so that they are done in the order of 0, 2, 4, 1, 3, and thus with p = 2 skipping.

    0 % 5 = 0
    2 % 5 = 2
    4 % 5 = 4
    6 % 5 = 1
    8 % 5 = 3 → all possible remainder have been cycled through
    10 % 5 = 0 → the cycle repeats

    I figure that someone must have recognized this and wrote it up as a lemma somewhere.
     
  2. jcsd
  3. May 17, 2016 #2

    FactChecker

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  4. May 17, 2016 #3

    fresh_42

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    Two integers ##p,q## are coprime if and only if there are integers ##n,m## such that ##np + mq = 1##.
     
  5. May 17, 2016 #4

    micromass

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    Yep, this is the answer. I just wanted to add that this is called Bezout's theorem. The specific integers ##n##, ##m## can be found very easily by the Euclidean algorithm.
     
  6. May 17, 2016 #5

    fresh_42

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    There's a name for it? I've always thought this is the first statement after the definition or the definition itself, sorry. It looks prettier with ideals: ##ℤ = pℤ + qℤ##
     
  7. May 17, 2016 #6

    micromass

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    You know the result for non coprime ##p## and ##q## too?
     
  8. May 17, 2016 #7

    fresh_42

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    What's the english name of it? Biggest common divisor? And which property of ℤ is essential? :smile:
     
  9. May 17, 2016 #8

    micromass

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    Greatest common divisor or gcd, as opposed to lcm the least common multiple.
     
  10. May 17, 2016 #9

    fresh_42

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    This would have been a real short answer: ℤ is a principal ideal domain.
     
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