While doing the leapfrog division for 13/97 it occurred to me that in this case using two digit division produces digits a lot faster.
This does not mean that there is something wrong with the leapfrog division. There isn’t. It just means that there are multiple ways of dividing and sometimes one is quicker and other times the other is quicker.
The procedure is dividing by 100 and then adding 3 times the digits produced to the remainder. This is the standard correction needed to go from dividing by 100 to 97.
Since we are multiplying a 2 digit number by 3, calculating the remainder is easy and should produce answers quickly.
Let’s revisit 13/97 doing this procedure.
1300/100=13. Remainder: 13x3 = 39. So 1300/97 is 13r39.
3900/100=39. 39x3 = 117. 117 is bigger than 97, so aad one to 39 to get 40 and change 117 to 20 (subtract 97) => 40r20. In other words, 39r117 = 40r20
Answer so far: 1340
2000/100=20. 20x3=60
Answer so far: 134020
6000/100=60r0. 60x3 = 180. 60r180 = 61r83
Answer so far: 13402061
8300/100 = 83. 83x3 = 249. 83r249 = 85r55
Answer so far: 1340206185
5500/100 = 55. 55x3 = 165. 55R165 = 56r68
Answer so far: 134020618556
6800/100 = 68. 68x3 = 204. 68r204 = 70r10
Answer so far: 13402061855670
1000/100 = 10. 10x3 = 30.
Answer so far: 1340206185567010
Same answer as we got here https://artofmemory.com/forums/dividing-by-full-reptend-primes#comment-34797, just half the amount of steps.
If you are new to this kind of division, visualize having to divide 1300 apples over 97 baskets. We divide them first over 100 baskets, putting 13 apples into each basket.
Then we realize we need only 97 instead of 100. So we take out the apples from the last 3 baskets - giving 3x13=39 apples and use them as remainder.
Now we have 97 baskets filled with 13 apples and a remainder of 39.
In other words:
13/97 = 13r39.
This is the first step we did above.