Let's dive into division with the elegant sutra: "Paravartya Yojayet," which means "Transpose and Apply." This is a fantastic method for dividing numbers when your divisor is slightly bigger than a base like 10, 100, or 1000.
Instead of complex division, we turn the problem into simple multiplication and addition! The key is to find how much "bigger" our divisor is than its base and then use the negative of that number.
- Setup: Divisor is 12 (Base 10). Surplus is 2. Transposed number is -2. Split the dividend 434 into quotient part 43 and remainder part 4.
- Step-by-Step:
- Bring down the first digit, 4. This is our first quotient digit.
- Multiply this 4 by -2: 4 × -2 = -8.
- Add this to the next digit (3): 3 + (-8) = -5. This is our second quotient digit.
- Our raw quotient is "4, -5", which means 4 tens - 5 ones = 35.
- Multiply the last quotient digit (-5) by -2: -5 × -2 = 10. Add this to the remainder part (4): 4 + 10 = 14.
- Adjustment: Raw result is Q:35, R:14. Since 14 is bigger than 12, we adjust! 14 ÷ 12 = 1 R 2. Add 1 to the quotient (35 + 1 = 36). The final remainder is 2.
- The Answer: Quotient is 36, Remainder is 2.