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“The 16 Sutras”

“The 16 Sutras”. Examples: 1) 25 2  = 2 X (2+1) / 25 = 625 2) 35 2 = 3 X (3+1) /25 = 3 X 4/  25 = 1225 3) 135 2 = 13 X 14/25 = 18225. Number Base 14 10      8 10 10 10 97 100 112 100 993 1000

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“The 16 Sutras”

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  1. “The 16 Sutras”

  2. Examples: 1) 252  = 2 X (2+1) / 25 = 625 2) 352 = 3 X (3+1) /25 = 3 X 4/  25 = 1225 3) 1352 = 13 X 14/25 = 18225

  3. Number Base 14 10      8 10 10 10 97 100 112 100 993 1000 1011 1000

  4. Example: 23                               x   13                          ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯                          2 : 6 + 3 : 9  =  299

  5. Example: Divide 1225 by 12.  12         122      5

  6. Example: Divide 1225 by 12.  12         122      5            - 2                    ¯¯¯¯¯    

  7. Example: Divide 1225 by 12.  12         122      5          - 2        -2               ¯¯¯¯¯     ¯¯¯¯    

  8. Example: Divide 1225 by 12.  12         122      5            - 2         -2           ¯¯¯¯     ¯¯¯¯                 10     

  9. Example: Divide 1225 by 12.  12         122      5            - 2         -20              ¯¯¯¯¯     ¯¯¯¯                 10     

  10. Example: Divide 1225 by 12.  12         122      5            - 2         -20              ¯¯¯¯¯     ¯¯¯¯                 102       

  11. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102       

  12. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1

  13. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore:

  14. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12

  15. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12 =

  16. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12 = 102

  17. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12 = 102 Remainder =

  18. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12 = 102 Remainder = 1

  19. Example 2:      5(x+1) = 3(x+1) Solution: ( x + 1 )              x + 1 = 0     gives     x = -1

  20. Example :                      3x +  7y = 2                       4x + 21y = 6 Observe that the y-coefficients are in the ratio 7 : 21 i.e., 1 : 3, which is same as the ratio of independent terms i.e., 2 : 6 i.e., 1 : 3. Hence the other variable x = 0 and 7y = 2 or 21y = 6 gives y = 2 / 7.

  21. Example 1:                            45x – 23y = 113                            23x – 45y = 91 add them,                i.e., ( 45x – 23y ) + ( 23x – 45y ) = 113 + 91                i.e., 68x – 68y = 204         x – y = 3subtract one from other,                i.e., ( 45x – 23y ) – ( 23x – 45y ) = 113 – 91                i.e., 22x + 22y = 22           x + y = 1      and repeat the same sutra, we get x = 2 and y = -1

  22. Example 2:     x3 + 8x2 + 17x + 10 = 0         We know ( x + 3 )3 = x3 + 9x2 + 27x + 27       So adding on the both sides, the term ( x2 + 10x + 17 ), we get             x3 + 8x2 + 17x + x2 + 10x + 17 = x2 + 10x + 17     i.e.,, x3 + 9x2 + 27x + 27 = x2 + 6x + 9 + 4x + 8                  i.e.,, ( x + 3 )3 = ( x + 3 )2 + 4 ( x + 3 ) – 4                      y3 = y2 + 4y – 4 for y = x + 3                         y = 1, 2, -2. •     Hence x = -2, -1, -5         .

  23. Example: 123Step 1: Subtract the nearest power of ten from the number: 12 - 10 = 2Step 2: Double this number and add the number being cubed: (2 x 2) + 12 = 16Step 3: Subtract from this number the power of ten from step one: 16 - 10 = 6Step 4: Multiply this number by the answer in step one: 2 x 6 = 12Step 5: Cube the answer in step one: 23 = 8Step 6: Since we're cubing a 2 digit number, add two zeros to the answer in step two: 1600Step 7: Add 1 zero to the answer in step four: 120Step 8: Add the answers in steps seven and eight to the answer in step five: 1600 + 120 + 8 = 1728Thus, 123 = 1728

  24. Example :  3x2 + 7xy + 2y2 + 11xz + 7yz + 6z2. Step (i):   Eliminate z and retain x, y; factorize                3x2 + 7xy + 2y2 = (3x + y) (x + 2y) Step (ii):   Eliminate y and retain x, z; factorize                3x2 + 11xz + 6z2 = (3x + 2z) (x + 3z) Step (iii):   Fill the gaps, the given expression                  = (3x + y + 2z) (x + 2y + 3z)

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