Remainder Principal and why the 11 Principal works

This is a continuation of the previous post about the 11 Principal. I'll start by explaining with 9, 99, 999, and beyond, then extend to base-1, base²-1, base³-1, and beyond. Finally, why the 11 principal works.

To find if 9 is a factor of a number, sum the individual digits and see if the sum is nine or has a factor of 9. 2781 would be 2+7+8+1=18, 18 divides evenly by 9, so 2781 divides evenly by 9. If you don't know if the sum is divides evenly by 9, apply the process to the sum to generate a new sum. 2781-> 18 (1+8) ->9

If the repeated sum doesn't equal 9, you can subtract the sum you have from the number that generated it to make a number that will divide evenly by 9.
2782->19->10->1
1 is the remainder of 2782/9
This works for any number divided by 9 to find the remainder.

Now lets extend this idea to numbers divided by 99. Lets start with 9173625. We break the numbers in pairs: 09 17 36 25. We sum the pairs: 9+17+36+25=87. 9173625/99 has a remainder of 87. If the sum is more than 2 digits, repeat the process of pairing and summing until you are left with 1 or 2 digits.

For 999, break the number into sets of 3. 9173625 is 009+173+625=807, the remainder of 9173625/999. This process can be extended for as many 9's as you like. This is the first half of this principal, the decimal part.

The second part extends to base-1, base²-1, and so on. In hexadecimal (base 16), this is f, ff, fff, and more. 1e5902/f is 1+e+5+9+0+2=1f->10->1 so the remainder 1. 1e5902/ff is 1e+59+02=78 the remainder. 1e5902/fff is 1e5+902=ae7 as the remainder. This works for all positive integer bases greater than or equal to 2.

So what does this have to do with the Eleven Principal? Let me introduce something you may have learned in school.
(base-1)(base+1)=base²-1 or (x-1)(x+1)=x²-1
With base+1 as 11 every time. The addition and subtraction rules I listed for 11 in the previous post covers up what really happens. Back to decimal.

You remember the remainder rule for 9? It works for finding if a number divides evenly by 3. If the remainder is 3 6 or 9, then it divides evenly by 3. For remainder rule for 99, its factors are 9 and 11, so you can use the remainder rule to look for even division by 11 by looking for a remainder of 11 22 33 44 55 66 77 88 or 99. The 11 rule I showed earlier where you sum twice and subtract the sums works by breaking the digits in pairs via a weird method. 1793 would be 1+9 and 7+3 and then the pairs are subtracted 10-10, subtracting the ones and tens places to try to find 0, 11, 22 or so on.

These are just some basic ideas to find easy factors, explianed in an overly complicated way. The remainder principal can be used to find harder multiples to calculate for such as dividing by 37, which is a factor of 111, which is a factor of 999.

Hope you found this informative, helpful, or fun. Have a nice day, and feel free to ask any questions. :)