linguistic:paraphrasing
Method of least absolute remainders

In another version of Euclid's algorithm, the quotient at each step is increased by one if the resulting negative remainder is smaller in magnitude than the typical positive remainder. Previously, the equation

assumed that | r | > r > 0. However, an alternative negative remainder e can be computed:

if r > 0 or

if r < 0.

If r is replaced by e. when | e | < | r |, then one gets a variant of Euclidean algorithm such that

at each step.

Leopold Kronecker has shown that this version requires the fewest steps of any version of Euclid's algorithm. More generally, it has been proven that, for every input numbers a and b, the number of steps is minimal if and only if q is chosen in order that | r k + 1 r k | < 1 φ ∼ 0.618, {\displaystyle \left|{\frac {r_{k+1}}{r_{k}}}\right|<{\frac {1}{\varphi }}\sim 0.618,} where φ {\displaystyle \varphi } is the golden ratio.