Is there a lemma named for this?

[swampwiz]

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.

 

[FactChecker]

I think it is a consequence of Theorem RP#2 in http://people.sju.edu/~pklingsb/rp.fta.pdf

 

[fresh_42]

Two integers $p,q$ are coprime if and only if there are integers $n,m$ such that $np + mq = 1$.

 

[micromass]

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.

 

[fresh_42]

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: $ℤ = pℤ + qℤ$

 

[micromass]

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?

 

[fresh_42]

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?

 

[micromass]

What's the english name of it? Biggest common divisor?

Greatest common divisor or gcd, as opposed to lcm the least common multiple.

 

[fresh_42]

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