Analysis of the long division algorithm in the Knuth book (Seminumerical algorithms) 1
Theorem B on page no. 272 of the book states that,


  if $v_{n-1}$ $\geqslant$ $\lfloor b/2 \rfloor$, then q ∈ [ q̂ - 2 , q̂ ].


Therefore, the algorithm is valid only when $v_{n-1}$ $\geqslant$ $\lfloor b/2 \rfloor$, which makes $v_{n-1} \neq 0$ obvious. To ensure this we need to multiply both u and v by a factor ${(= (base-1) / v_{n-1})}$.