When it says "compute the greatest multiple of 98765432 that's less than or equal to 123456789", it actually means an
integer multiple. In other words, the division is turned into multiple
integer divisions, which are easy because they compute the result digit by digit, thus only allowing for 1 of 10 possible digits per step.
With big divisors, it makes sense to write down the first 9 multiples:
*1 98765432
*2 197530864
*3 296296296
*4 395061728
*5 493827160
*6 592592592
*7 691358024
*8 790123456
*9 888888888
With this, you no longer need to "compute" the greatest usable multiple, instead you simply check by looking at the numbers.
So now you can do the long division, which now requires nothing more than comparisons and subtractions:
123456789:98765432 = 1.2499...
98765432 <------/ ||||
-------- ||||
24691357 ||||
197530864 <--------/|||
--------- |||
49382706 |||
395061728 <---------/||
--------- ||
98765332 ||
888888888 <----------/|
--------- |
98764432 |
888888888 <-----------/
...
Of course, the subtraction result can never be 98765432 or over (otherwise you chose the wrong multiple). So at some point the subtraction result repeats, meaning you found the repeating part of the decimal.