Solve the following system of conguenees X 5mol 7 X 3mol 5 X

Solve the following system of conguenees X 5(mol 7) X 3(mol 5) X 2(mol 8) X 2(mol 3)

Solution

a ) x 5 mod 7

Let x = 5 + 7k ,k Z ,Z = {....-2,-1,0,1,2,...}

when k = -2 , x = 5 -14 = -9

   k = -1 , x = 5 - 7 = -2

k = 0 , x = 5 +0 = 5

   k = 1 , x = 5 + 7 = 12

   k = 2 x = 5 + 14 = 19

Therefore,

x = { .....-9,-2,5,12,19......}

b) x 3 mod 5

Let x = 3 + 5k ,k Z ,Z = {....-2,-1,0,1,2,...}

when k = -2 , x = -7

   k = -1 , x = -2

  k = 0 , x = 3

   k = 1 , x = 8

   k = 2, x = 13

Therefore,

x = { .....-7,-2,3,8,13......}

c)

  x 2 mod 8

Let x = 2+ 8k ,k Z ,Z = {....-2,-1,0,1,2,...}

when k = -2 , x = -14

   k = -1 , x = -6

  k = 0 , x = 2

   k = 1 , x = 10

   k = 2 , x = 18

Therefore,

x = { .....-14,-6,2,10,18......}

d)

  x 2 mod 3

Let x = 2 + 3k ,k Z ,Z = {....-2,-1,0,1,2,...}

when k = -2 , x = -4

   k = -1 , x = -1

  k = 0 , x = 2

   k = 1 , x = 5

   k = 2 , x = 8

Therefore,

x = { .....-4,-1,2,5,8......}

The above 4 lists have nothing in common.

Thus, there are no simultaneous solutions to the given 4 congruences.