Derive an algorithm to mentally divide any base ten (decimal) integer from 1 to 18 (inclusively) by 19.
There is the "divide by 2" method.
You keep dividing the numerator by two, keeping track of the integer quotient (iq) and remainder (r) and combining them as r*10 + iq for the next division by 2:
e.g.
9/19 =
9/2
r=1 iq=4
etc....
r iq
------
1 4, next divide 14 by 2 to get 7 with remainder 0:
0 7
1 3
1 6
0 8
....
Reading down the 2nd column is the answer: 0.47368...
The proof uses the expansion of 1/(1-epsilon)
n/(20-1) = (n/20) (1/(1-e)), with e=1/20
1/(1-e) = 1+1/e+1/e^2+... and with e=(1/10)(1/2), this is
n/19 = n (1/10 1/2 + 1/100 1/4 + 1/1000 1/8 ...)
(I saw this in a Presh Talwalker video years ago).
What is cool is that you can use this trick for dividing by 29 (now divide by 3s) or for 39 (divide by 4s), etc. e.g.
8/29 = 0.27586
8/3=
r iq
2 2
1 7
2 5
1 8
0 6
Edited on December 5, 2021, 11:35 pm
soln