GCD{$\ \cdot\ $}LCM = PROD for Rims:If {$T\in\mathcal {B}^+$} is a rim, then {$$\text{GCD}_{_T}\ \cdot\ \text{LCM}_{_T} = \text{PROD}_{_T}.$$} Proof: For rims, {$\text{GCD}_{_T}=(z_{_T},z_{_T},z_{_T})$}. Consider first the case when {$y_{_T}=z_{_T}=1$}, which means, by Relatively Prime=Loopless, that there are no loops. As before, this gives us {$$(L^1_{_T}+L^2_{_T}+L^3_{_T})=\text{PROD}_{_T}.$$} Since {$\text{GCD}_{_T}=(1,1)$}, we have {$\text{GCD}_{_T}\cdot\text{LCM}_{_T}=\text{PROD}_{_T}.$} The case for {$y_{_T}>1$} proceeds exactly as in GCD{$\ \cdot\ $}LCM = PROD for axles. |