Multiplication and Division with Vedic Mathematics

$321729$not divisible by $2,3,6$

We have,

Given that:

$587\times 999=587\times (1000-1)$
$=587\times 1000-587\times 1$
$=587000-587$
$=586413$

$93521\times 625=935421\times {5}^{4}=935421\times {(\cfrac{10}{2})}^{4}$
$\cfrac{935421\times{10}^{4}}{{2}^{4}}=\cfrac{9354210000}{16}$
$=584638125$

$2056\times 987=2056\times (1000-13)$
$=2056\times 1000-2056\times 13$
$=2056000-26728$
$=2029272$.

$72519\times 9999=72519\times(10000-1)$
$=72519\times 10000-72519\times1$
$=725190000-72519$
$=725117481$

$12345679\times 72=123456789\times (70+2)$
$=12345679\times 70+123445679\times 2$
$=864197530+24691358$
$=888888888$

By hit and trial , we find that
$47619\times=333333$.

$(112\times {5}^{4})=112\times { \left( \cfrac { 10 }{ 2 }  \right)  }^{ 4 }=\cfrac { 112\times { 10 }^{ 4 } }{ { 2 }^{ 4 } } =\cfrac { 1120000 }{ 16 } =70000$

Required numbers are $10,15,20,25,......96$
This is an A.P. in which $a=10,d=5$ and $l=95$
${t} _{n}=95$ $\Rightarrow$ $a+(n-1)d=95$
$\Rightarrow$ $10+(n-1)\times 5=95$
$\Rightarrow$ $(n-1)\times 5=85$
$\Rightarrow$ $(n-1)=17$
$\Rightarrow$ $n=18$
$\therefore$ Required Sum $=\cfrac { n }{ 2 } (a+l)=\cfrac { 18 }{ 2 } \times (10+95)=(95\times 105)=945$

$5358\times 51=5358\times (50+1)$
$=5358\times 50+5358\times 1$
$=267900+5358$
$=273258$

1397×1397=13972.

78 * 76 = 5928. Using the Nikhilam formula with a base of 80 (sub-base 8), the calculation is (80-2) * (80-4) = 5928.

132 * 124 = 16368. Using the Urdhva Tiryakbhyam method, you multiply vertically and crosswise to obtain the product.

57 * 68 = 3876. The Urdhva Tiryakbhyam method involves multiplying digits vertically and crosswise.

349 * 986 = 344114. This is calculated using the Urdhva Tiryakbhyam method for 3-digit multiplication.

41 * 41 = 1681. This is a standard square calculation.

12 * 14 = 168. This is a standard multiplication.

234 * 316 = 73944. This is calculated using the Urdhva Tiryakbhyam method.

123 * 45 = 5535. This is calculated using the Urdhva Tiryakbhyam method.

32 * 24 = 768. This is calculated using the Urdhva Tiryakbhyam method.

390 * 11 = 4290. Multiplying by 11 can be done by shifting and adding.

9999  × 9
= (100000 - 1) 9
= 99999 - 9
= 89991

987 * 11 = 10857. Multiplying by 11 is a standard Vedic math technique.

111 * 11 = 1221. This is a standard multiplication.

28 * 22 = 616. Using the Nikhilam formula or standard multiplication, the result is 616.

432  × 9
= 432  × (10 - 1)
= 4320 - 432
= 3888

386 * 11 = 4246. Multiplying by 11 is a standard Vedic math technique.

999 ×  11 = 999 (10 + 1)
= 9990 + 999
= 10989

Using Nikhilam with base 100, the deviations are -1 and -4. The product is (99-4) and (-1*-4), which is 95 and 04, resulting in 9504.

10001  × 99 = 10001 (100 - 1)
= 1000100 - 10001
= 990099

90098  × 99 = 90098 (100 - 1)
= 9009800 - 90098
= 8919702

5555  × 9 = 5555 (10 - 1)
= 55550 - 5555
= 49995

1111  × 9 = 1111 (10 - 1)
= 11110 - 1111
= 9999

9999  × 99 = 9999 (100 - 1)
= 999900 - 9999

1111  × 9 = 1111 (10 - 1) = 11110 - 1111

9999  × 99 = 9999 (100 - 1)
= 999900 - 9999
= 989901

12345  × 99 = 12345 (100 - 1)
= 1234500 - 12345

12345  × 99 = 12345  × (100  - 1)
= 1234500 - 12345
= 1222155

9999  × 9 = 9999 (10 - 1) = 99990 - 9999

For 103^3, the base is 100 and deviation is +3. The formula is (number + 2deviation) | (3deviation^2) | (deviation^3), which gives (103 + 6) | (3*9) | 27 = 109 | 27 | 27 = 1092727.

The deviations from base 100 are -2, -3, and -1. The product is calculated by (98-3-1) | ((-2*-3) + (-3*-1) + (-1*-2)) | (-2*-3*-1), which is 94 | (6+3+2) | -6. Adjusting the -6 gives 94 | 11 | -6 = 94 | 10 | 94 = 941094.

The deviations from base 100 are -2, -1, and -1. The calculation is (98-1-1) | ((-2*-1) + (-1*-1) + (-1*-2)) | (-2*-1*-1), which is 96 | (2+1+2) | -2 = 96 | 5 | -2 = 964498.

The two rightmost digits in the multiplication 98,97, 99 = 100 - (2 + 3 + 1) = 100 - (2)(3)(1)

Using base 50 (sub-base 510), the deviations are +1, +2, +3. The calculation is 5^2 * (51+2+3) | 5(12 + 23 + 31) | (123) = 2556 | 5*11 | 6 = 1400 | 55 | 6 = 140556.

The two rightmost digits in the multiplication 98,97, 99 = 100 - (2 + 3 + 1) = 100 - 6 = 94

99  × 99 = 99 (100 - 1)
= 9900 - 99
= 9801

The correct representation of deviations from the base in the multiplication of 101,102,103 by Nikhilam formula of vedic maths is 01, 02 and 03.

For 12/9 with base 10, the divisor 9 has a complement of 1. The division 12/9 gives quotient 1 and remainder 3.

For 96^3, the base is 100 and deviation is -4. The formula is (n + 2d) | 3d^2 | d^3 = (96 - 8) | 3(16) | (-64) = 88 | 48 | -64. Adjusting the negative remainder: 88 | 47 | (100-64) = 884736.

Total number of voters $= 21,35,680$
Number of polling booth $= 235$
So, number of voters at each polling booth $= 21,35,680 \div 235 = 9088$.

Dhwajanka method calculations: 3987/28 gives Q=142, R=11; 5786/78 gives Q=74, R=14; 7396/82 gives Q=90, R=16.

46 * 4 = 184. The vinculum method represents 46 as 5(bar 4), and the multiplication follows standard arithmetic.

Given that $31z5$ is a multiple of $9$. 

Using sub-base 50, deviations are +4 and +7. Product = 5 * (54+7) | (4*7) = 5 * 61 | 28 = 305 | 28 = 3078.

Using sub-base 60, deviations are +3 and -2. Product = 6 * (63-2) | (3*-2) = 6 * 61 | -6 = 366 | -6 = 3654.

Using sub-base 10, deviations are +1 and +5. Product = 1 * (11+5) | (1*5) = 16 | 5 = 165.

Deviations from 10 are +2, +3, +7. Product = (12+3+7) | (23 + 37 + 72) | (23*7) = 22 | (6+21+14) | 42 = 22 | 41 | 42 = 2652.

$(102 \times 98) -(9.5 \times 10.5)=(100 + 2) (100 -2) -[(10 -0.5) (10 + 0.5)]$
$=(100^2-2^2) -(10^2 -0.5^2)$
$=10000 -4 -(100 -0.25)$
$=10000 -104 + 0.25$
$=9896 + 0.25=9896.25$

Urdhva Tiryakbhyam for 349 * 986 involves cross-multiplication groups: (39) | (38+49) | (36+48+99) | (46+98) | (9*6).

Using base 10, deviations are +2 and +8. Product = (12+8) | (2*8) = 20 | 16 = 216.

For 65^3, using sub-base 60 (deviation +5). Formula: 6^2 * (65+2*5) | 6 * 3 * 5^2 | 5^3 = 36 * 75 | 18 * 25 | 125 = 2700 | 450 | 125 = 274625.

For 96^3, the base is 100, so the deviation is 96 - 100 = -04.

In the Nikhilam cube calculation for 96, the last part is the cube of the deviation (-4)^3 = -64. To make this positive, we borrow from the previous column, resulting in 100 - 64.

For 422/11, base 10, divisor 11 has a deviation of 1. Using Nikhilam division, 422 / 11 = 38 with remainder 4.

Deviations from 10 are -1, -2, +5. Product = (9-2+5) | ((-1*-2) + (-25) + (5-1)) | (-1*-2*5) = 12 | (2-10-5) | 10 = 12 | -13 | 10 = 1080.

Deviations from 100 are +1, +2, +3. Product = (101+2+3) | (12 + 23 + 31) | (12*3) = 106 | 11 | 06 = 1061106.

80^3 = 8^3 * 10^3 = 512 * 1000 = 512000.

Using sub-base 30, deviations are +2 and +8. Product = 3 * (32+8) | (2*8) = 3 * 40 | 16 = 1200 + 16 = 1216.

Cost of $1$ liter milk = $Rs.19.75$

Using sub-base 50, deviations are +1 and +5. Product = 5 * (51+5) | (1*5) = 5 * 56 | 05 = 280 | 05 = 2805.

Length of board $=\text{7 feet 9 inches}$

The cubage (cube) of 71 using the Nikhilam formula involves base 70. 71^3 = (71 + 21) * 70^2 + 31^2 * 70 + 1^3 = 73 * 4900 + 210 + 1 = 357700 + 211 = 357911.