Let us start our proceedings with the help of three examples.
Examples of division
 Find remainder when 1245 is divided by 4
 Multiply 5 by 1. The result is 5.
 Take 1 and multiply by 10. The result is 10. Find remainder when 10 is divided by 4. The result is 2.
 Multiply 4 by the result of step 2. 4×2 = 8.
 Take 2, multiply by 10. The result is 20 and find remainder when 20 is divided by 4. The result is 0.
 Calculate the sum of result of step 1 and 3. 5+8 = 13.
 As the result of step 4 is zero so there is no need to calculate further.
 Find remainder when 13 is divided by 4. Result is 1. This is the result when 1245 is divided by 4.
 Find remainder when 1245 is divided by 3
 Multiply 5 by 1. The result is 5.
 Take 1 and multiply by 10. The result is 10. Find remainder when 10 is divided by 3. The result is 1.
 Multiply 4 by the result of step 2. 4×1 = 4.
 Take 1, multiply by 10. The result is 10 and find remainder when 10 is divided by 3. The result is 1.
 Multiply 2 by the result of step 4. 2×1 = 2.
 We find that the remainder in step 2 and 4 has repeated. So the remainder will be same in all the step afterwards.
 Multiply 1 by 1. 1×1 = 1.
 Calculate the sum of result of step 1, 3, 5 and 7. 5+4+2+1 = 12.
 Find remainder when 12 is divided by 3. Result is 0. This is the result when 1245 is divided by 3.
 Find remainder when 1245 is divided by 7
 Multiply 5 by 1. The result is 5.
 Take 1 and multiply by 10. The result is 10. Find remainder when 10 is divided by 7. The result is 3.
 Multiply 4 by the result of step 2. 4×3 = 12.
 Take 3, multiply by 10. The result is 30 and find remainder when 30 is divided by 7. The result is 2.
 Multiply 2 by the result of step 4. 2×2 = 4.
 Take 2, multiply by 10. The result is 20 and find remainder when 20 is divided by 7. The result is 6.
 Multiply 1 by 6. 1×6 = 6.
 Calculate the sum of result of step 1, 3, 5 and 7. 5+12+4+6 = 27.
 Find remainder when 27 is divided by 7. Result is 6. This is the result when 1245 is divided by 7.
In the above examples we find remainder which we multiply with each digit of the number. These remainders are given here.
Remainders Sequences.
We see different sequences for different numbers. Some sequences terminate some not. Let us see an example of sequence for the number 19.
Divisibility by 19
Method 1:
The number is divisible by 19 if the number follows the following method and the remainder is zero.
Let us find the remainder of 42653876 when divided by 19.
Remember the sequence 1, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2, 1, 10, 5, 12, 6,....
Here 1, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2 repeat in the sequence.
0.Reverse the order of
digits 
6 
7 
8 
3 
5 
6 
2 
4 
1.Multiply with
the sequence 
6×1 
7×10 
8×5 
3×12 
5×6 
6×3 
2×11 
4×15 
 
6 
70 
40 
36 
30 
18 
22 
60 
2.Add the numbers 
6+70+40+36+30+18+22+60=282 
3.Repeat step 0 with
the added number 
2 
8 
2 
0 
0 
0 
0 
0 
4.Multiply with
the sequence 
2×1 
8×10 
2×5 
0 
0 
0 
0 
0 
 
2 
80 
10 
0 
0 
0 
0 
0 
5.Add all 
2+80+10 =92

7.Divide the result by 19
find the remainder 
mod(92/19)=16

If the remainder is zero (0) then the number is divisible by 19 else the result is remainder.
The remainder is 16. Repeat the step 0,1 and 2 till you get a number whose remainder you can find easily.
Method 2:
The number is divisible by 19 if the number follows the following method and the remainder is zero.
Let us find the remainder of 42653876 when divided by 19.
Remember the sequence 1, 9, 5, 7, 6, 3, 8, 4, 2, 1, 9, 5, 7, 6, 3, 8, 4, 2, 1, 9, 5, 7, 6,....
Here 1, 9, 5, 7, 6, 3, 8, 4, 2, 1, 9, 5, 7, 6, 3, 8, 4, 2 repeat in the sequence. Here 1, 9, 5, 7, 6, 3, 8, 4, 2 is followed by negative of them in order.
0.Reverse the order of
digits 
6 
7 
8 
3 
5 
6 
2 
4 
1.Multiply with
the sequence 
6×1 
7×9 
8×5 
3×7 
5×6 
6×3 
2×8 
4×4 
 
6 
63 
40 
21 
30 
18 
16 
16 
2.Add the numbers 
663+4021+30+181616= 22 
Remainder= mod(22/19) =3
Real Remainder = 193 = 16 
If the remainder is zero (0) then the number is divisible by 19 else the result is remainder.
The remainder is 16. Repeat the step 0,1 and 2 till you get a number whose remainder you can find easily.
Remainder
 In method 1 we can surely say that the result is the remainder of the original number.
 In method 2 we are not sure. If the result is <19 and >0 then the number is the remainder else 19+(negative number) is the remainder. 18≤negative number≤1.
To have a look at some methods for other numbers Read
two (2) ,
three (3) ,
four (4) ,
five (5) ,
six (6) ,
seven (7) ,
eight (8) ,
nine (9) ,
ten (10) ,
eleven (11) ,
twelve (12) ,
thirteen (13) ,
fourteen (14) ,
fifteen (15) ,
sixteen (16) ,
seventeen (17) ,
eighteen (18) ,
nineteen (19) ,
twenty (20) ,
twenty one (21) ,
twenty two (22) ,
twenty three (23) ,
twenty four (24) ,
twenty five (25) ,
Reason for sequence to work well for remainders
The sequence are formed of the remainders when the place value is divided by the divisor. The remainder is for one. The remainder for a different number x will be x times of this remainder. The number which is composed of place value and face value with their product can be expressed as a number, divisor and remainder. This thing is known to everyone. Now the remainder becomes x times for a value x. All the remainders can be added and checked for remainder.