Is there a lemma named for this?

  • #1
285
10

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
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  • #3
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Two integers ##p,q## are coprime if and only if there are integers ##n,m## such that ##np + mq = 1##.
 
  • #4
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Two integers ##p,q## are coprime if and only if there are integers ##n,m## such that ##np + mq = 1##.
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.
 
  • #5
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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.
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##
 
  • #6
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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##
You know the result for non coprime ##p## and ##q## too?
 
  • #7
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You know the result for non coprime ##p## and ##q## too?
What's the english name of it? Biggest common divisor? And which property of ℤ is essential? :smile:
 
  • #8
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What's the english name of it? Biggest common divisor?
Greatest common divisor or gcd, as opposed to lcm the least common multiple.
 
  • #9
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This would have been a real short answer: ℤ is a principal ideal domain.
 

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