True for {$T'$} Claim:We claim that the proposition holds for the box {$T'=\langle a_{_T},b_{_T},c_{_T}\rangle$}. Proof: We can see that on {$\langle a_{_T},b_{_T},c_{_T}\rangle$} there are eight paths of length {$a_{_T}+b_{_T}$},  and four paths of length {$b_{_T}$},  so {$\text{LCM}_{_{T'}}=(a_{_T}+b_{_T},a_{_T}+b_{_T},b_{_T})$}. This gives us {$$\text{GCD}_{_T}\cdot\text{LCM}_{_T}=(b_{_T}-a_{_T},a_{_T},a_{_T})\cdot(a_{_T}+b_{_T},a_{_T}+b_{_T},b_{_T})=(b_{_T}^2-a_{_T}^2)+(a_{_T}^2+a_{_T} b_{_T})+(a_{_T} b_{_T})=b_{_T}^2+2a_{_T} b_{_T},$$} which is the same as {$\text{PROD}_{_{T'}}$}. |