Technique explanation
To divide by 1 + d (where d is small), use the Paravartya Yojayet (Transpose and Adjust) method with -d as the multiplier. Separate the remainder digits equal to the number of digits in d. Bring down the first quotient digit, multiply by the transposed digits, and add to subsequent columns.
This method is best for divisors where digits after the leading '1' are small. Standard fallback: Conventional long division.
Worked examples
Example 1: 1225 divided by 12
- Divisor: 12. Transposed digit: -2.
- Setup: 1 | 2 | 2 || 5. (One digit for remainder).
- Step 1: Bring down 1. 1 (-2) = -2. 2 + (-2) = 0.
- Step 2: Bring down 0. 0 (-2) = 0. 2 + 0 = 2.
- Step 3: Bring down 2. 2 (-2) = -4. 5 + (-4) = 1.
- Result: Quotient 102, Remainder 1.
- Check: (102 × 12) + 1 = 1224 + 1 = 1225.
Example 2: 13456 divided by 113
- Divisor: 113. Transposed digits: -1, -3.
- Setup: 1 | 3 | 4 || 5 | 6. (Two digits for remainder).
- Step 1: Bring down 1. 1 (-1, -3) = -1, -3. Col 2: 3-1=2. Col 3: 4-3=1.
- Step 2: Bring down 2. 2 (-1, -3) = -2, -6. Col 3: 1-2=-1. Col 4: 5-6=-1.
- Step 3: Bring down -1. -1 (-1, -3) = 1, 3. Col 4: -1+1=0. Col 5: 6+3=9.
- Normalize: 1, 2, -1 -> 119. Remainder 0, 9 -> 9.
- Result: Quotient 119, Remainder 9.
- Check: (119 × 113) + 9 = 13447 + 9 = 13456.
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