
Steps  97 ÷ 12  
1.  Find a Base, that is near the Divisor  that is a power of 10, that is 10^{n}.  In this case, the close power of 10 that is nearest to 12 is 10, which is the Base.  
2.  Find the Transposition of the Divisor  as (Base  Divisor), and represent as Vinculum, if required. Make sure that the Transposition has the same number of digits, as the Zeroes of the Base, 10^{n}  In this case, the transposition is: 10  12 = 2 = 2 

3.  Arrange the numbers in a manner, that the Divisor is slightly raised, from its Transposition, and the Dividend on the righthand side.  In this case, ^{12} 2 ) 97 

4.  Leave some space between the digits of the Dividend, and partition it into 2 parts: 1st Part is for Quotient, and the 2nd Part is for Remainder, such that the digits for the Remainder should be same  as the Zeroes of the Base, 10^{n}. 
In this case, ^{12} 2 ) 9  7   

5.  Drop the first digit (or the cummulative sum of any other number in the same column) of the Dividend, and multiply it (the 'Dropped' digit)with each digits of the Transposition  placing them in rows, next to the place of the 'Dropped' digit.  In this case, the first digit is dropped: ^{12} 2 ) 9  7   9  And, Multiplying the dropped digit, with each digits of the Transposition: ^{12} 2 ) 9  7  18  9  Note: 9 × 2 = 9 × 2 = 18 = 18 

5.  Drop the next digit (or the cummulative sum of any other number in the same column) of the Dividend, and repeat above step (Step 4) till all the places for every digit of the Dividend is exhausted.  In this case, the next digit is dropped: ^{12} 2 ) 9  7  18  9  11 All places for the digits in the Dividend is exhausted, so we stop. Note: 7 + (18) = 11 = 11 

6.  Finalize the Quotient and Remainder parts as: 1. Remove any viniculum number from Quotient; 2. Remove any viniculum number from Remainder; 3. If the Remainder is negative, take appropriate numbers of Quotient to make it positive; 4. If the Remainder is not less than the Dividor, give appropriate numbers to Quotient to make it lesser. 
In this case, The Quotient part contains: 9 And, the Remainder part contains: 11 = 11 The Remainder is negative. So, we take one from the Quotient: 9  1 = 8 And give it to Remainder: 1 × 12  11 = 1 So, we get: Quotient = 8, and Remainder = 1, which is the answer! 
The Base is 1000 and, 1000  1212 = 212 = 212 So, we arrange as: ^{1212} 212 ) 1 0  4 0 6     Step 1  First Digit: After dropping the first digit, and multiplying it with all the digits of the Transposition, we get: ^{1212} 212 ) 1 0  4 0 6 2  1 2   1  Note above, that 1 × 2 = 2, 1 × 1 = 1, and 1 × 2 = 2. And, it is arranged in next row, leaving the digit that is dropped. Step 2  Next digit: We drop 0 + 2 = 0 + (2) = 2 = 2, and multiplying it with all the digits of the Transposition, we get: ^{1212} 212 ) 1 0  4 0 6 2  1 2  4 2 4  1 2  Note above, 2 × 2 = 4, 2 × 1 = 2, and 2 × 2 = 4. And, it is arranged in next row, leaving the digit that is dropped. Step 3  Next digit: All the places of the Dividend is exhausted, so we stop  after totalling the columns: ^{1212} 212 ) 1 0  4 0 6 2  1 2  4 2 4  1 2  7 0 10 Note above, (4 + 1 + 4) = 4  1 + 4 = 7, (0 + 2 + 2) = 0  2 + 2 = 0, and (6 + 4) = 10 So, we get: Quotient = [1,2] = 10  2 = 8 And, Remainder = [7,0,10] = 710 Since the Remainder is positive and less than the Quotient, no further finalizing is required. Thus, 10406 ÷ 1212 = 8, and Remainder: 710 
The Base is 1000 and, 1000  1013 = 13 = 013 (The number of digits of the Transposition, should be the number of Zeroes of the Base) Working ahead, we get: ^{1013} 013 ) 1 3 1  6 9 0 0 1  3 0  3 9  0 0 0  1 3 0  0 0 0 So, we get: Quotient = [1,3,0] = 130 And, Remainder = [0,0,0] = 0 Since the Remainder is Zero, 131690 is divisible by 1013. Thus, 131690 ÷ 1013 = 130 
The Base is 100 and, 100  98 = 2 = 02 (The number of digits of the Transposition, should be the number of Zeroes of the Base) Working ahead, we get: ^{98} 02 ) 2 8 2  3 7 0 4  0  16  0 12  2 8 6  19 19 So, we get: Quotient = [2,8,6] = 286 And, Remainder = [19,19] = 209 But 209 is greater than 98. If we give one to Quotient, the Remainder becomes 209  98 = 111 This is still greater than 98, so we give another to Quotient, and the Remainder becomes 111  98 = 13 Now, we have our final Quotient = 286 + 2 = 288, and Remainder = 13 Thus, 28237 ÷ 98 = 288, and Remainder: 13 Note that, not even a single change needs to be incorporated for a number less than the Base. 

28237 ÷ 98 = 288, and Remainder: 13 (We have achieved this, in the above example) Working ahead with the Remainder, we get: ^{98} 02 ) 1 3 0  0 0 0 2  0  6  0 4  1 3 2  6 4 Note above, that the imaginary Zeroes are greyedout. So, we get 13/98 as: Quotient = [1,3,2] = 132 And, Remainder = [6,4] = 64 But, we suffixed 3 Zeroes, so the Quotient has to be divided by 1000, which becomes 0.132 Thus, 28237 ÷ 98 = 288.132... Note that, more decimal point may be derived by suffixing more imaginary Zeroes. Also, the method remains fairly simple if the Remainder is obtained, and then later reworked for decimal points  rather than trying to obtain the Quotient and decimal figures at the same time. 

This division method is selfexplanatory, because it is only a variation of the 'polynomial long division' method, where the Transposition is only a representation of (x  r), where x is Base and r is the Divisor. Not only does this tactic provide a marked advantage by reducing the division to multiplications of digits  it is also a generalized tactic that will work with any ndigit Divisor. 
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