Find $ k$ last digits of quotient Suppose we have two integer numbers $a, b$ such that $b$ divides $a.$ Suppose  that  the number $a$ is very and very long  and we cant to perform a division algorith. How to find last $k$ digits of the quotient $\frac{a}{b}$ for small $k?$
I know how  to find last digits for the  product $a b$ (just cut the last $k$  numbers of $a$ and $b$)  but have any simple idea for their quotient.
 A: I guess you mean k decimal digits. If $\gcd(b,10^k)=1$ you can use your product algorithm together with modular arithmetic to compute $a\times c \bmod 10^k$ with $c=b^{-1} \bmod 10^m$ with $m>k$. Example
$$b=1234567, k=10, m=14, d=170141183460469231731687303715884105733$$ 
$$a=b \times d=210050690441241118011293999486607892762472611$$
With these values you have $c=21861510958903$ and $a\times c= \dots 3715884105733.$
where you can read-off the last $10$ digits of $d$ and get $5884105733$
Other possibilities are the Newton method, or Jebelean’s exact division algorithm
(see the GMP manual).
A: Since the question is tagged [algorithms], and this is not e.g. stackoverflow but rather the theoretical world of mathematics, there is an algorithm that doesn't use division at all, only multiplications as requested...

$j \gets 10^k$
  $i \gets 0$
  repeat
  $\quad$if $i \times b = a$,
  $\quad\quad$ take $(i \bmod j)$ as solution
  $\quad$fi
  $\quad i \gets i + 1$
  taeper

$\text{O}(\frac{a}{b})$ time complexity.
