Is long division the most optimal "effective procedure" for doing division problems? (Motivation: I am going to be working with a high school student next week on long division, which is a subject I strongly dislike.)
Consider:  $\frac{1110}{56}=19\frac{46}{56}$.
This is really a super easy problem, since once you realize $56*20=1120$ its trivial to write out $1110=56*19+46$.
You can work out the long division for yourself if you want; needless to say it makes an otherwise trivial problem into a tedious, multi-step process.
Long division is an "effective procedure", in the sense that a Turing machine could do any division problem once it's given the instructions for the long division procedure.  To put it another way, an effective procedure is one for which given any problem of a specific type, I can apply this procedure systematically to this type of problem, and always arrive at a correct solution.
Here are my questions:
1) Are there other distinct effective procedures for doing division problems besides long division?
2) Is there a way to measure how efficient a given effective procedure is for doing division problems?
3) Does there exist an optimal effective procedure for division problems, in the sense that this procedure is the most efficient?

In binary computing multiplying by $2$ is a shift. But humans also like to double numbers.
An efficient algorithm can be constructed that only multiplies by $2$, subtracts numbers, compares numbers and adds numbers.
Here is this division algorithm applied to $\frac{1110}{56}$.
Create the table:
$56 \times 2^0 \quad \;\, \quad \quad\quad = \;\, 56;\quad 2^0 = 1$
$56 \times 2^1 = 56 \times 2 \;\,= 112;\quad 2^1 = 2$
$56 \times 2^2 = 112 \times 2 = 224;\quad 2^2 = 4$
$56 \times 2^3 = 224 \times 2 = 448;\quad 2^3 = 8$
$56 \times 2^4 = 448 \times 2 = 896;\quad 2^4 = 16$
$56 \times 2^5 = 896 \times 2 \gt 1110 \text{ == STOP == }$
From this point on all that is needed to complete the calculation are the subtraction, comparison and addition operators.
$1110 = 56 \, 2^4 + (1110 - 896) = 56 \, 2^4 + 214 = 56 \, 2^4  +56 \, 2^1 + (214 - 112) = $
$\quad 56 \, 2^4  +56 \, 2^1 + 102 = 56 \, 2^4  + 56 \, 2^1 + 56 \, 2^0 + (102-56) =$
$\quad 56 \, 2^4  + 56 \, 2^1 + 56 \, 2^0 + 46 =$
$\quad (2^4  + 2^1 + 2^0)\,56 + 46 =$
$\quad (16 + 2 + 1)\,56 + 46 =$
$\quad 19\times 56 + 46 $
