Is there a lemma named for this?

  • #1
swampwiz
445
35
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

  • #3
fresh_42
Mentor
Insights Author
2021 Award
16,935
16,634
Two integers ##p,q## are coprime if and only if there are integers ##n,m## such that ##np + mq = 1##.
 
  • #4
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,129
3,302
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
fresh_42
Mentor
Insights Author
2021 Award
16,935
16,634
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
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,129
3,302
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
fresh_42
Mentor
Insights Author
2021 Award
16,935
16,634
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
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,129
3,302
What's the english name of it? Biggest common divisor?

Greatest common divisor or gcd, as opposed to lcm the least common multiple.
 
  • #9
fresh_42
Mentor
Insights Author
2021 Award
16,935
16,634
This would have been a real short answer: ℤ is a principal ideal domain.
 

Suggested for: Is there a lemma named for this?

  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
11
Views
5K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
883
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
5
Views
2K
Replies
2
Views
12K
Replies
2
Views
788
  • Last Post
Replies
2
Views
1K
Top